华东师范大学数学系：常微分方程课后答案（中山大学王高雄版）：odexxzd

x0x1x2x3x4x8x5x9xa
x0x1x2x3x4x5x6x5x7
2004x81x9
x8x9
xaxbxc xdxexfx10x11x12x13x14x15
1.1 x0x1x2x3x20x7x8x9xa 1
1.2 xbxcxdxe 4
1.3 x7x8xfx10x11x12 20
xax16xc x17x18x19x1ax1bx1cx1d
2.1 x13x10 26
2.2 x14x15xfx10 30
2.3 x16x17x18x19x1ax0x1x2x3x1b 36
2.4 x1cx1dx19x1ax0x1x2x3 43
xax1exc x1fx18x11x20x1fx18x21x22
3.1 x7x8x9xa 54
3.2 x1ex1dx17x1fx20x21x1ax1x22 58
3.3 x14x15x23x19x1ax17x1fxex3ex6ax20x25x21x1a 64
3.3 x7fx59 69
1
x41x42x43 x29x2ax2bx2cx2dx2ex2fx30x31
x1.1 x1cx1dx19x1ax38x21x22x23x24
x4x9x7fx52x0x1
1,x32x11x12x13x14 x24x25x32x11x12x13x14x2x3x4x5x7bx6x5x7x22x8x4x5x5fx5cx5dx9xa
xbx49x23x46xexfx60x11x12x7bxaxbx49x23x60x10x23x11x4ax77x22x60x12x13.
x17x77x24x9x6x72x8x74x75x76x33x2x7cx7dxc
dx
dt = x
2 +1; (1:1:1)
d2x
dt2 = sint; (1:1:2)
d2x
dt2 = 1; (1:1:3)
xdx+tdt = 0; (1:1:4)
x58x49(1.1.1)x1bx60x7ex25x20x26t,(1.1.2)x1bx60x7ex7fx7bx22x23x,(1.1.3)x1bx60x7ex25
x20x26tx1fx7fx7bx22x23x,(1.1.4)x9x8x74x75x76x33x2x74x75x36x64,x58x49x20x26tx1fxx2ax79
x2ax36x25x20x26x1fx27x20x26x3cx30x37x38x4dx28x29x2ax20x26x52x13x2x63x2fx3cx28x2ax63x2fx1bx6x36x9x3e
x2bx2x22x23x63x2f.
x10x2ax73x1fx49x1ax1bx9x2x2dx29x6dx60“x8x74x75x76x33”x27“x74x75x76x33”x23x69x6dx60x27“x76
x33”.
x1ax74x75x76x33x49,x58x36x7ex79x7fx7bx22x23x2x73x23,x58x49x5ax3cx2xex25xax23x24x60x27x77
x11x12x13x14x60x19x23;x74x75x76x33x49x4fx17x1bx60x7ex25x20x26x69x7fx7bx22x23x10x25.
x6x64nxax8x74x75x76x33x4fx71x72x71x77x2cx76x33x36x64
F(t;x; dxdt;:::; d
nx
dtn ) = 0; (1:1:5)
x58x49F x9x58x20x67x2x7ax7bx22x23.
x3dx1ax45x46x8x8x34x10xex25xax73x23x7ax9x5ax2x1ex7dx1dx13
dnx
dtn = f(t;x;
dx
dt;:::;
dn?1x
dtn?1 ); (1:1:6)
x73x76x33x2x0x68x9x7fx7bx22x23x2xex25xax73x23 (n xax73x23),x55x76x33x2x67x68x27x25x20x26x42
x7fx7bx22x23x1fx7fx7bx22x23x53x5cnxax2x73x23x2x7ax7bx22x23.
1
x59 x1 = x,x4fx17x6a n xax76x33 (1.1.6) x2x27x71x77 n x2ax7fx7bx22x23 x1,x2,:::,
xn x2nx2ax6xax76x33x2bx4bx34x10xc
8
>>>>
>>>
>>>
<
>>>>
>>>
>>>
:
dx1
dt = x2;dx
2
dt = x3;.
..
dxn?1
dt = xn;dx
n
dt = f(t;x1;:::;xn);
(1:1:7)
x4fx17x6ax30x2x27x6x2ax77x26(x10x1fx49x7bx2ex3fx2fx0x72xbx77x26)x2x74x75x76x33:
dx
dt = f(t;x); (1:1:8)
x58x49x = (x1;:::;xn)T,f = (0;:::;f)T x5dx27nx66x21x77x26.
2,x22x11x7ax23x22x11 x40nxax74x75x76x33(1.1.5)x49x2x22x23F x63x5cx7fx7bx22x23x,
x17xcx7fx7bx22x23x2x73x23 dxdt,:::,d
nx
dtn x2x27n+1x2ax20x26x2x15x3fx9x6x5x79x39x15x64,x7cx7dx24x60x30x27nx19x22x11x13x14,x30x2x6x64x36x64x9
a0(t)x(n) +a1(t)x(n?1) +￠￠￠+an?1(t)x0 +an(t)x = f(t); (1:1:9)
x58x49 a0(t) 6· 0,x40 f(t) · 0,x8x60x76x33 (1.1.9) x27x3ax3bx6ax5x76x33,x40 f(t) 6· 0,
x8x60x76x33 (1.1.9) x27x3ax3bx39x6ax5x76x33,x40 a0(t) 6= 0 x29x3cx4fx62x66x64x29x68x57x6bx17
a0(t)x24x6cx4dxex25xax73x23xdx2x2fx23x91x2x33x30x36x64,x22x23x77x26x2x6xax3ax3bx74x75
x76x33x2x6x64x36x64x27
dx
dt = A(t)x+f(t); (1:1:10)
x58x49x,f x5dx27nx66x21x77x26,x55A(t)x27nxax76x25(x10x1fx49x17x7bx28x69x2fx3fx72xbx6b
x25,x2ex3fx2fx72xbx77x26).
x1bx9x3ax3bx2x76x33x24x60x27x23x22x11x13x14,x7cx71x0 = sinx,xx0 = t,x0 = x2,x17
x9x39x3ax3bx76x33,x27x27 sinx x1bx9 x x2x6x5x22x23; xx0 x63x5cx7fx7bx22x23xcx58x73x23x9
x1ax5x2x3c x2 x63x5cx7fx7bx22x23x9x1ax5x2,x55 x0 = t2 x9x3ax3bx76x33x3cx27x27 t2 x9x7ax7b
x22x23x3cx63x5cx7fx7bx22x23x1fx58x73x23x9x6x5x2.
3,x11x12x13x14x60x2a x71x13x6ax7ax7bx22x23 x = ’(t) x69x22x23x77x26x = ’(t) xcx58
x73x22x23x64x65x2cxdx2x74x75x76x33,x51x6cx77x74x75x76x33x1ax22x23’(t)x69x22x23x77x26’(t)x2
x36x7x5dx13I x66x72x27x7fx8x64,x8x60x28x69x22x23’(t)x69x22x23x77x26’(t)x27x74x75x76x33x1a
x5dx13I x66x2(x60x64)x9,x28x2ax5dx13I x60x27x74x75x76x33x2x9x2x36x7x5dx13,x57x6x2ax74x75
x76x33x4fx17x79x1bx57x2x9,x1bx57x2x9x2x36x7x5dx13x4fx17x1bx57.
x11x36x7x4fx50x9x22x23x2x36x7x5dx13I x2x5x30x6x36x23x5cx2,x1bx67x9x22x23x2x73x23x24
x2ax79x74x7,x36x7x5dx13x4fx50x9x20x5dx13,x7fx5dx13,x69x71x20x71x7fx5d.
x40x76x33x2x9x9x11x2cx22x23x4dx36x2,x8x60x52x27x2cx64x9,x40x76x33x2x9x9x11x46x23x36
x64x72xbx2,x8x60x52x27x46x23x36x64x9.
2
4,x2dx2ax9x7dx2ax24x25x7ax2ex2a x3x5c n xax76x33 (1.1.5) x3bx7e,x71x13x1ax30x2x9
x = ’(t;c1;:::;cn) x49x7ex79 n x2ax2fx4x2x14x74x8x23 c1;:::;cn,x8x60x28x2ax9x27
(1.1.5)x2x2dx2a.
x7cx71dx=dt = 2tx2x27x9x27x = t2+c,x3dy = x2+c1+c2x49x2x8x23c1x1fc2
x24x1bx9x2fx4x2,x27x27x4fx17x62x28x29x2ax8x23x2cx14x55x72x27x6x2ax8x23,x2bx6cx73x74x2x9x2d
x2ax32x4x7dx26x33x34x13x14x35x2ax60x2a;x7cx71x4fx17x4dxdx = ct?c2 x9x76x33x = tx0?x02
x2x27x9,x58x49cx27x14x74x8x23;x28x2ax76x33x78x79x6x2ax9x = t2=4x24x1bx3ax3bx1ax30x2x27
x9x49.
x11x5cx74x75x76x33x2x27x9x19x7ex79x14x74x8x23,x27x63x27x4dx4ex36x30x2x28x2ax44x36x2x9,x78
x58x3fx6x5ax77x9x24xdx37x38x2x13x6b,x28x69x13x6bx24x60x27x7dx2ax7cx40,x36x9x13x6bx11x5cx45x46
x75x76x2x1bx57x79x71x69x71x58,x7cx7dx1ax10x1fx49x56x6ex6fx7x21x13x6b,x7x21x13x6bx9x2ax6dx25x20
x26x1ax28x6x6x36x4x29,x7fx7bx22x23x17xcx30x2x53x5cx76x33xax23x2x73x22x23x1ax77x4xdx16x6x36
x2x23x2b,x3x5cx76x33(1.1.5)x2x7x21x13x6bx27
x(t0) = x0; x0(t0) = x1;,:,x(n?1)(t0) = xn?1;
x58x49 x0;x1;:::;xn?1 x27x6x36x2x7ax7bx23x2b,x55x3x5cx77x26x76x33 (1.1.7) x2x7x21x13
x6bx27
x(t0) = x0;
x28x28t0 x27x25x20x26 tx1ax58x20x21x5dx13I x49x2x6x2ax6x36x4,x55x0 x27x6x36x2nx66x8
x77x26.
x7cx7dx6ax74x75x76x33x2x1bx7ex14x74x8x23x2x9x60x27x77x76x33x2x2ex2a.
x7cx71x4fx17x4dxd x = c1 cost + c2 sint x9x1axax76x33 x00 + x = 0 x2x27x9,x27
x63x1bx4fx59x5ax00+x = 0x37x38x7x21x13x6bx(0) = 1,x0(0) = 0x2x44x9x27x = cost.
x17x9x4x19x13x14x2ax60x6x4fx50x51
x3x5cx6xax74x75x76x33,x7x30x66x7bx,y x2x27x20x26x3cx5fx30x8x8x7bxx2x27x25x20x26x3c
y x2x27x7fx7bx22x23.
1,x79x12x52x22 x52 D x27 x;y x5cx6cx66x2x5dx30,x6ex6fx74x75x76x33
dy
dx = f(x;y); (x;y) 2 D,(1:1:11)
x52x22x23 y = ’(x) x27 (1.1.11) x2x6x2ax44x9,x28x22x23x1a x;y x5cx6cx66x2xax7bx9 D
x49x2x6x13x53x3a,x7cx7dx60x30x27x76x33(1.1.11)x2x6x13x79x12x52x22,x52(1.1.11)x2x27x9
x9,y = ’(x;c),x6dx58x49x14x74x8x23 c x1ax28x6x45x23x18x0x49x16x2bx29,x24x6cx4d D x49x2
x40x75x53x3ax2x18x0,x7cx7dx60x30x27x79x12x52x22x55,(1.1.11) x37x38x7x21x13x6b y(x0) = y0
x2x44x9x24x9x27x32x4 (x0;y0) 2 D x2x40x75x53x3a,x63x6,x1a (1.1.11) x2x40x75x53x3a
x66x14x6x4 (x;’(x)) x5,x58x45x3ax56x6 ’0(x) x31x3dx8x5cx22x23 f(x;y) x1ax77x4x5x2
x2b f(x;’(x)),x73 y = ’(x) x37x38x76x33 (1.1.11) ; x32x52,x71x13x3x5c D x49x2x14
x6x13x57x58x53x3a y = ’(x),x1ax30x66x6cx14x6x4x5x2x45x3ax56x6 ’0(x) x59x3dx24x9x22x23
f(x;y) x1ax77x4x5x2x2b f(x;’(x)),x8x63x53x3ax24x9(1.1.11)x2x40x75x53x3a.
3
x1.2 x25x26x57x27
x10x3x62xexfx71x15x62x28x72x6xax76x33x2x9x7bx7x8x22x23x69x7bx7x8x22x23x2x40x75x3bx72
xb,x28x69x9x7x60x27x7x8x9x7,x66x67x28x72x76x33x2xbx1x3cx79x6f,x3dx3x5cx45x46x47x48x49
x5ax3cx2x8x74x75x76x33x4ax7x8x7bx4d,x27x63x70x71x28x72xbx1x76x33x2x9x7x78x9x79x3fx6fx2x45
x46x42x2bx2.
x6x42x12x72x5cx5dx73
1,x5cx5dx12x72x13x14 x7cx7dx60x4fx71x72x36x71
dy
dx = f(x)g(y) (1:2:1)
x2x6xax76x33x27x5cx5dx12x72x13x14,x58x49f(x) x1f g(y)x75x9x27 x,y x2x6ex6fx22x23,x7cx71
x76x33y0 = y2 cosx,y0 = ex+y,y0 =
p1+y2
p1?x2 x17x9x20x26x75x17x76x33.
x27x4dx59x9(1.2.1),x79x7ax51x52 g(y) 6= 0,x5cx9(1.2.1)x4fx71x72
dy
g(y) = f(x)dx;
x28x29x3cx1ax76x33x49x3dx6x2axdx56x7ex79x6x2ax20x26x3cx28x2ax32x33x60x27x20x26“x75x17”,x29x68x40
x75x73x6cx2cx64x27x9 Z
dy
g(y) =
Z
f(x)dx+c; (1:2:2)
x28x28xcx17xdx6bx39x44x45x75x76x7cx7dx6ax1bx36x40x75
Z dy
g(y) x1f
R f(x)dx x75x9x39x9x27
y x1f x x2x6x2a(x1bx74x14x74x8x23x2)x2bx22x23,x30x7dx1bx6x36x9x7x8x22x23,x55 c x9x51
x6c(1.2.2) x79x74x7x2x14x74x8x23,x7cx71x76x33 dydx = sinxxy,x7x75x17x20x26xdx40x75x4fx6c
y2 = 2
Z sinx
x dx+c;x66x67x67x1x1bx50x7bx7x8x22x23x72xb,x3dx7cx7dx75x27x27x9x7ax7x59
x5a.x2dx71x76x33 dydx =?xy,x28x29(1.2.2)x27 y2 =?x2 +c,x60x67x28x29 c x56x50x16x39
x5ax2x14x74x45x23,x19x52,(1.2.2)x24x9(1.2.1)x2x27x9.
x58x5,x71x13x3ax1a y0 x51x6c g(y0) = 0,x8x7x53x54x4dxdx73x7b y = y0 x65x9(1.2.1)
x2x9,x28x29x71x13x1ax27x9 (1.2.2) x49x3ax1ax8x23 c = c0 x51x6cx30x3ax7ex4dx9 y = y0,
x8x27x9(1.2.2)x24x3ax7ex4d(1.2.1)x2x6x45x9,x79x8(1.2.1)x2x9x6bx4dx27x9(1.2.2)
x6,x78xdx14x66x9 y = y0.
x76 1.2.1 x59x9x76x33 dydx = y2 cosx,x5fx59x37x38x7x21x13x6b y(0) = 1 x2x9.
x2a x60x67y = 0x9x6x2ax9x3cx6dy 6= 0x29,x4fx75x17x20x26x6c
dy
y2 = cosxdx; (x52y 6= 0)
4
x29x68x40x75x73x6c
1y = sinx+c:
x27x63x27x9x27
y =? 1sinx+c,(1:2:3)
x28x28 c x27x14x74x8x23,x3dx9x4bx10 c x16x77x58x2x8x23x2b,x17x1bx9x9x9 y = 0,x27x55x2b
x76x33x2x6x45x9xdx11x66x38x27x9x1fx44x9y = 0xex72.
x27x4dx59x5ax7x2bx47x48x2x9,x79x7ax1dx5ax9 y = 0 x60x67x1bx37x38x7x21x13x6b,x27x63x56
x50x1ax27x9x49x4ax4c.x27x63x1a(1.2.3)x49x26 x = 0,y = 1 x73x6c c =?1,x5cx9x24x59x7
x2bx47x48x2x9x27 y = 11?sinx.
x76 1.2.2 x59x3ax3bx6ax5x76x33
dy
dx = p(x)y (1:2:4)
x2x27x9,x58x49 p(x) x27 x x2x6ex6fx22x23.
x2a x60x67 y = 0x9(1.2.4)x2x9,x40y 6= 0,x76x33x29x68x24x17y?1,x6c
y?1dy = p(x)dx;
x29x68x40x75x6cx4dx2cx64x27x9
ln(y=c) =
Z
p(x)dx;
x58x49 c x27x39x2x2x14x74x45x8x23,x8x64x67x68x72xb p(x) x2x14x6x2ax2bx22x23,x11x3x23x3b
x43x4fx21x27x60x64x27x9
y = cexp
Z
p(x)dx
,(1:2:5)
x28x28xcx17xdx8x7b exp(t) x3bx72xbx2ax23x22x23 et,x71x13x16x3e (1.2.5) x49 c = 0x3cx8
(1.2.5) x3ax7ex4dx44x9 y = 0,x27x63 (1.2.5) x4fx17x72xb (1.2.4) x2x6x45x9,x56x6fx75x27
x58x49cx27x14x74x45x8x23.
x3x5cx74x75x36x64x2x20x26x75x17x76x33:
f(x)g(y)dx+h(y)k(x)dy = 0;
x58x49f(x),g(y),h(y),k(x)x17x9x7ax7bx22x23,k(x)g(y) 6= 0x29x3cx7cx7dx4fx17x62x76x33
x2x29x68x57x6bx17k(x)g(y)x3bx75x17x20x26x3cx6c
f(x)dx
k(x) +
h(y)dy
g(y) = 0:
x29x68x40x75x6cx27x9,Z
f(x)dx
k(x) +
Z h(y)dy
g(y) = c:
5
x40x79x0 x69y0,x51k(x0) = 0x69g(y0) = 0,x8x76x33x78x79x44x9x = x0,x69y = y0.
x1ax74x75x36x64x2x74x75x76x33x49x2ax79x19x36x36x6x2ax9x25x20x26x3cx36x6x2ax9x7fx7bx22x23.
x27x63x74x75x36x64x2x76x33x2x40x75x53x3ax9 x;y x5cx6cx66x2x53x3ax3cx30x4fx50x1bx9x4x5x22x23
x2xax7b,x71x66x48x49x2x44x9 x = x0 x24x1bx9x6x2ax17 x x27x25x20x26x2x22x23,x55x9x6x2a
x17y x27x25x20x26x2x22x23,x71x20x26x75x17x76x33
2xdx+2ydy = 0
x29x68x40x75xdx6cx4dx2x2cx64x9x9x2+y2 = c2 > 0,x30x2xax7bx9x71x4cx27c > 0,x5bx1bx1a
x2bx4x2x5b,x30x6fx7bx29x2ax60x64x3ex2bx22x23x3bx72xbxc y = §(c2?x2)1=2; x 2 [?c;c].
2,x78x61x18x5cx5dx12x72x60x13x14
1– x7cx7dx60x4fx71x72x36x71
dy
dx = g
y
x
·
(1:2:6)
x2x76x33x27x79x7ax13x14(x27g x27x,y x2x2x5x6ax5x22x23x55x6cx69,x28x2ax6ax5x76x33x2x3x4
x4ex29x6cx3ax3bx6ax5x76x33x2x3x4x1bx57,x31x1bx6fx2x2d,x6ax5x22x23x2x36x7xcx3bx43x50x7x48
x18),x58x49g(u)x27ux2x6ex6fx22x23,x7cx71x76x33y0 = y=x+ (y=x)2,y0 = 2xyx2?y2,
(x2+y2)dx+(xy?y2)dy = 0 x17x9x6ax5x76x33,xexdx6x2ax76x33x6dy(y?x) 6= 0
x29x4ex76x33,y0 = (x2 +y2)=(y2?xy) x9x8x42x2.
x59x9x6ax5x76x33x2x63x7bx9x3x7fx7bx22x23x2bx4bx20x26x7cx7d,x5ex73x6ax22x23 f x2x20x26
y=xx7bx6x2ax7ex2x7fx7bx22x23ux64x7c,x24x4fx62(1.2.6)x21x72x20x26x75x17x76x33,x5cx9
y = xu; (1:2:7)
x52u x27 x x2x22x23,x5cx9
dy
dx = u+x
du
dx; (1:2:8)
x62(1.2.6)x49x67x68x2y=xx7bux64x7cxdx1bx64x65(1.2.8)x2x0x68x73x6c
g(u) = u+xdudx;
x11x63x70x5a
du
dx =
g(u)?u
x,
x28x9x6x2ax20x26x75x17x76x33,x6dg(u)?u 6= 0x29x3cx79x27x9
Z du
g(u)?u = ln(cx);
x1bx26u = y=xx4fx6cx2bx76x33(1.2.6)x2x27x9,x28x29x40x3ax1a u0 x51x6c g(u0)?u0 = 0,
x8x78x79x44x9,y = u0x x 6= 0.
x76 1.2.3 x59x9x76x33 dydx = yx +tan yx.
6
x9 x26 y = xu,x62x63xc y0 = u+xu0 x64x65x2bx76x33x6c
u+tanu = u+xu0;
x5ex73
xu0 = tanu:
x6dtanu 6= 0x29,x4fx21x27x63x5cx7fx7bx22x23siny x2x3ax3bx6ax5x76x33
dsinu
dx =
1
x sinu;
x11x37x64(1.2.5)x6c
sinu = cexp
Z dx
x
= cx:
x58x49 c x27x14x74x8x23,x6a u = y=x x64x65x6cx2bx76x33x2x27x9 sin(y=x) = cx,x78x79x44
x9y = k…x; k 2 Z;x 6= 0,x9x11tanu = 0x6cx4d.
2– x71x77x22x11x12x13x13x14
dy
dx =
a1x+b1y +c1
a2x+b2y +c2; a
2
2 +b
2
2 6= 0; (1:2:9)
x65x4fx17x59x7bx20x26x7cx7dx21x72x20x26x75x17x76x33,x58x49 a1;a2;b1;b2;c1;c2 x5dx27x8x23.
(i) x6d c1 = c2 = 0 x29,x62(1.2.9)x67x1x2x75x7dx75x0x57x6bx17 x,x73x6c(1.2.6)
x36x64x2x76x33.
(ii) x6d a1 = a2k,b1 = b2k x29,x8(1.2.9)x4fx71x72
dy
dx =
k(a2x+b2y)+c1
a2x+b2y +c2 =,f(a2x+b2y);
x5cx9x26u = a2x+b2y x52xd,x2bx76x33(1.2.9)x72x27
du
dx = a2 +b2
dy
dx = a2 +b2f(u);
x60x67x28x9x6x2ax20x26x75x17x76x33.
x1ax10x1fx49,x2bx7bx59x2d“:=”x1f“=:”,x30x7dx2x7ex7x9x36x7,A,= B x72xb“x36x7
x59x2dAx27x72x3ex64B”,x65x4fx71x72B =,A.
(iii) x6dx1bx9x66x38x29x69x75x36x29,x3ax3bx64x23x76x33xe
‰ a
1x+b1y +c1 = 0;
a2x+b2y +c2 = 0
x3ax1axax6x9 x = fi;y = fl,x5cx9x26
x =? +fi; y = · +fl;
x8x76x33(1.2.9)x72x27
dy
dx =
d·
d? =
a1? +b1·
a2? +b2·;
7
x28x24x9x75x76(i).
x63x6,x1bx4fxdx7fx66x38x9x48x76x7x65x1x7bx5cx1a(1.2.9)x2fx6x64x2x71x77x76x33
dy
dx = f
a
1x+b1y +c1
a2x+b2y +c2
:
3– x2x3x4x1d x77x21x76x33x2x59x9x5x6x1cx1dx2x27x6x7xc
dy
dx = f(ax+by +c); x26u = ax+by +c;
yf(xy)dx+xg(xy)dy = 0; x26u = xy;
x2dydx = f(xy); x26u = xy;
dy
dx = xf
y
x2
·; x26u = y=x2;
M(x;y)(xdx+ydy)+N(x;y)(xdy?ydx) = 0;
x58x49 M,N x27 x,y x2x5x23x4fx17x1bx57x2x6ax5x22x23,x59x7bx8x32x33x20x7dx = rcos,
y = rsin x52xdx2bx4bx20x26x75x17x59x9.
x76 1.2.4 x59x9x76x33
dy
dx =
x?y +1
x+y?3,(1:2:10)
x2a x79x7ax59x5ax3ax3bx64x23x76x33xe
‰ x?y +1 = 0;
x+y?3 = 0
x2x9 x = 1;y = 2,x67xdx26 x =? +1;y = · +2;x62x63x64x65(1.2.10)x6cx6ax5x76
x33
d·
d? =
·
+· =
1?·=?
1+·=?,(1:2:11)
x1bx26 · =?u,x8(1.2.11)x21x72
u+?dud? = 1?u1+u
x2bx6x74x21x27
du
d? =?
u2 +2u?1
(1+u)? ;
x22x74x75x6cx7fx7bx22x23(u2 +2u?1)x2x3ax3bx6ax5x76x33
d(u2 +2u?1)
d? =?2
u2 +2u?1;
8
x11x37x64(1.2.5)x6c
u2 +2u?1 = cexp
Z
2d?=?
= cexpln?2￠ = c2:
x68x4dx20x26(?,·)x73x6c(1.2.11)x2x27x9
·2 +2?·2 = c;
x1bx68x4dx20x26(x,y),x73x7b(1.2.10)x2x27x9x27
y2 +2xy?x2?6y?2x = c;
x58x49cx27x14x74x8x23.
x73xcx11x5cx10x48x2x44x45x3bx3cx79x6x69x6dx3ex2x9x7xcx62x76x33(1.2.10)x5ex71x72x71x77
x74x75x36x64:
(xdy +ydx)+[(y?3)dy?(x+1)dx] = 0;
x2bx55x4fx71x72xc
d(xy)+[(y?3)d(y?3)?(x+1)d(x+1)] = 0;
x40x75x6fx4x6cx27x9,x28x69x76x7x60x27x75xex40x75x7x3cx56x50x7bx5cx6x72x44x45x2x76x33x9
x1ax42x22x11x13x14x46x32x23x5cxax73
x6d(1.1.9)x49x2 n = 1 x29,x7cx7dx6cx4dx4x19x22x11x11x12x13x14x2x6x64x36x64
a0(x)y0 +a1(x)y = f(x):
x16xdx7cx7dx19x51x52 a0(x) 6= 0,x27x27x6d a0(x) x5ax3cx2x4x29,x66x6cx76x33x24x72x27x74x75
x76x33x9x71x39x10x26x27x2x3x35,x27x63x7cx7dx19x9x6ax6xax3ax3bx76x33x71x72x71x77x2x33x30x36x64
(x50(1.1.6) )
dy
dx = p(x)y +q(x); (1:2:12)
x58x49x51x52 p(x),q(x) x1ax24x34x10x2x5dx13x66x27 x x2x6ex6fx22x23.
x6dq(x) · 0x29,(1.2.12)x72x27(1.2.4)
dy
dx = p(x)y,(1:2:4)
x7cx7dx60(1.2.4)x27x4x19x22x11x79x7ax13x14,x60x30x9x3xdx5cx39x6ax5x76x33(1.2.12)x2
x6ax5x76x33,x55x6dq(x) 6· 0 x29x60(1.2.12)x27x4x19x22x11x23x79x7ax13x14,x60q(x) x27x3a
x3bx76x33(1.2.12)x2x23x79x7axb.
x3ax3bx6ax5x76x33(1.2.4)x2x9,x7cx7dx7ax1ax66x6cx2x7c1.2.2x49x34x10x32,x58x27x9x27
y = cexp
Z
p(x)dx
,(1:2:5)
x28x28 c x27x14x74x8x23.
9
x11x5c (1.2.4) x9 (1.2.12) x2x44x45x75x36,x27x63x4fx17x52x3a (1.2.12) x2x27x9xdx6d
x9 (1.2.5) x2x28x69x6ex6fx4d q(x) x2x70xc; x55x28x69x70xc (1.2.5) x2x6x2ax25x67x2x4cx7
x24x9x6ax14x74x8x23 c x20x6dx27 x x2x6ax36x22x23 c(x),x51x6cx30x37x38(1.2.12),x59h(x)
x27x3xdx5c(1.2.12)x2x6ax5x76x33x2x14x6x39x2x9x3cx3cx59(1.2.12)x71x77x36x64x2x9
y = c(x)h(x); (1:2:13)
x62(1.2.13)x64x65(1.2.12)x6c
dc(x)
dx h(x)+c(x)
dh(x)
dx = c(x)p(x)h(x)+q(x):
xfx6dh(x)x9(1.2.4)x2x9,x66x64x49x7ex79c(x)x2xdx4fx3ax4ax3cx21x6dx27
dc(x)
dx =
q(x)
h(x);
x29x68x3 x x40x75x73x6c
c(x) = C +
Z q(x)
h(x)dx; (1:2:14)
x58x49C x27x14x74x8x23,x62(1.2.14)x64x65(1.2.5)x73x6c(1.2.12)x2x27x9xc
y = h(x)
C +
Z q(x)
h(x)dx
,(1:2:15)
x66x6cx28x69x62x3ax3bx6ax5x76x33x27x9x49x2x14x74x8x23x20x6dx27x6ax36x22x23,x57x55x59x5ax3a
x3bx39x6ax5x76x33x27x9x2x76x7,x7cx7dx60x27x32x23x5cxax73,x17xdx7cx7dx78x62x3fx5x7bx4d.
x63x6x1bx4fx53x54x4dxd,x76x33(1.2.12)x37x38x7x21x13x6b y(x0) = y0 x2x9x27
y(x) = exp
Z x
x0
p(t)dt

y0 +
Z x
x0
q(t)exp
Z t
x0
p(s)ds
dt
:
x76 1.2.5 x59x76x33 y0 = 2xy +exp(x2)cosx x2x27x9.
x2a x79x7ax59x7bx37x64(1.25),x59x3xdx6ax5x76x33 y0 = 2xy x2x39x2x9h(x):
h(x) = exp?x2￠:
x58x5x59x7bx37x64(1.2.15)x6cx2bx76x33x2x27x9
y = exp?x2￠
C +
Z
cosxdx
= exp?x2￠(C +sinx); (1:2:16)
x58x49C x27x14x74x8x23.
x76 1.2.6 x59x5ax76x33 dydx = yx+y3 x2x27x9.
x2a x60x67x28x2ax76x33x63x5c y x9x39x3ax3bx2,x30x1bx50x2bx4bx20x26x75x17,x3dx6dy 6= 0
x29,x4fx6ax30x71x72 dxdy = x+y
3
y,x73
dx
dy =
1
yx+y
2;
10
x6axx1dx72x7fx7bx22x23,x55x6ay x1dx72x25x20x26,x28x24x9x6x2ax3ax3bx76x33,x53x54x59x7bx37x64
(1.2.15),x16x3xdx2x6ax5x76x33x2x9h(y) = exp?R dy=t￠ = y,x4fx6cx2bx76x33x2x27x9
x71x77xc
x = y
C +
Z
ydy
= y
C + 12y2
:
x78x79x44x9y = 0.
x73,x1ax59x6ax5x76x33x2x39x2x9 h(x) x29x3cx79x29x9x2bx4dx40x75x71 exp?R adx=x￠
xdx77x16x79x19x3x2bx2x2bx22x23jxja,x78x9x16x1bx74x19x3x2bx2x2bx22x23xax2x47x48.x45x46x66
x4fx17x2bx16x6x2ax2bx8,x6d x < 0 x29 xa x79x74x7x29x1bx14x19x3x2b,x27x27x28x29 xa x6bx67
x9x6ax5x76x33x2,x55x1bx14x19x3x2bx3x5cx59x39x6ax5x76x33x2x44x9x3bx7ex6fx76x6fx6x72,x3dx6d
x < 0x29xa x4bx74x7x29x58x3fx6fx14x19x3x2b.
xexdx1axex38x10x1fx52x29,x7cx7dx78x6ex6fx6xbx4fx27x32x20x26x7cx7dx55x21x72x3ax3bx76x33x2b
x4bx59x9x2x76x33,x30x79x71x77x36x64xc
dy
dx = p(x)y +q(x)y
n; (1:2:17)
x58x49 p(x);q(x) x27 x x2x6ex6fx22x23,n 6= 0;1 x27x45x8x23,x7cx7dx60x76x33 (1.2.17)
x27xfx10x11 (Bernoulli) x13x14,x65x9x6xbx8x50x2x76x33,x27x4dx59x9 (1.2.17),x76x33x2
x29x68x57x6bx17yn,(x40n > 0,x6fx59y 6= 0)),x6c
y?ndydx = p(x)y1?n +q(x)
x4fx50x40x7cx7dx6ay1?n x2x27x6x2ax7ex2x7fx7bx22x23,x73x26
u = y1?n; (1:2:18)
x5cx9x79
du
dx = (1?n)y
ndy
dx,(1:2:19)
x62(1.2.17)x64x65(1.2.19)x2x67x68,x5fx6ex6fx4d(1.2.18)x73x6c
du
dx = (1?n)p(x)u+(1?n)q(x):
x28x9x6x2ax63x5cx7fx7bx22x23 u x2x3ax3bx76x33,x11 (1.2.15) x73x6cx30x2x27x9,x67xdx11
(1.2.18) x68x4dx2bx3bx20x26,x6fx6c (1.2.17) x2x27x9,x63x6x6d n > 0 x29,(1.2.17)
x78x79x9 y = 0.
x76 1.2.7 x59x5ax76x33 xy0 +y = (xlnx)y2 x2x27x9.
x2a x6dx50x76x33x79x6x44x9x27 y = 0,x6d y 6= 0 x29x3cx26 u = 1y,x8x79 dudx =
y?2dydx,x5cx9x2bx76x33x20x72x63x5cx7fx7bx22x23ux2x3ax3bx76x33
du
dx =
1
xu?lnx;
11
x11(1.2.15)x4fx6cx63x76x33x2x27x9x27
u = x
C? 12 ln2 x;
x68x4dx20x26y,x73x6cx2bx76x33x2x27x9x27
xy
C? 12 ln2 x
= 1:
x4ax6x2ax7x8x2bx4dx2x76x33x9x12x13x14(Riccati)x13x14
dy
dx = p(x)y +q(x)y
2 +f(x);
x58x49 p(x),q(x),f(x) 6· 0 x17x9xx2x6ex6fx22x23,x60x67x40f(x) · 0,x8x30x24x9x15
x16x59x76x33,x55x6d f(x) 6· 0x29,x6x64x2ax79x7x8x59x9x2x76x7; x3dx71x13x7ax7bx30x2x6x2a
x44x9 y = ’(x),x8x4fx27x32x26 y = u?1 + ’(x) x55x6cx4dx6x2ax63x5c u x2x3ax3bx76x33,
x57x55x4fx59x5ax30x2x27x9,x73x74x3cx28x2ax44x9x6x64x1bx3ax7ex1ax27x9x49.
x65x42x18x11x12x13x14x46x79x12x5bx19
1,x18x11x12x13x14x7cx7dx6ex6fx71x77x74x75x36x64x2x76x33
M(x;y)dx+N(x;y)dy = 0; (1:2:20)
x58x49M,N x27x,y x2x6ex6fx4fx74x22x23,x28x58x2x36x64x79x29x2fx6fx5cx4bx4cx30x2x27x9.
x6dx76x33(1.2.20)x2x0x1x62x3dx9x28x2ax1ax67x22x23u(x;y)x2x4cx74x75x29,x5ex73
M(x;y)dx+N(x;y)dy · du(x;y) = @u@xdx+ @u@ydy; (1:2:21)
x24x60 (1.2.20) x27x18x11x12x13x14x2dx60x27x15x6c (exact)x13x14,x40 (1.2.20) x27x4cx74x75x76
x33,x8x58x27x9x27x2cx22x23x36x64
u(x;y) = c:
x28x28 c x27x14x74x8x23,x28x69x36x64x2x27x9x60x27x74x75x76x33x2x40x75,x27x9x49x2x3dx2ax9x1a
x;yx5cx6cx66xax7bx65x60x27x74x75x76x33(1.2.20)x2x40x75x53x3a,x3dx9x28x2ax40x75x53x3ax4fx50
x6fx7bx40x4bx2ax17 x x69 y x27x25x20x26x2x60x22x23x3bx72xb,x4fx9x6x64x7cx7dx1bx58x57x2cx22x23
x59x5ax60x64x9x3b,x7cx71 ydx+xdy = 0x9x4cx74x75x76x33,x79d(xy) = 0,x12x58x27x9x27
xy = c,x6d c = 0 x29,x40x75x53x3ax24x79x29x13,x = 0 x1f y = 0,x4fx17x1dx72x75x9x9x17
y x1fxx27x25x20x26x2x22x23.
x3cx1ax50x1ax7cx7dx6cx29x2x79x65x2ax47x48xc
1– x71x15x61x1c(1.2.20)x9x79x27x4cx74x75x76x33x78
2– x40(1.2.20)x27x4cx74x75x76x33,x71x15x59x5ax22x23 u(x;y)x78
3– x40(1.2.20)x1bx9x4cx74x75x76x33,x79x15x4cx7x62x30x1bx20x72x4cx74x75x76x33,x5fx59x5a
x58x27x9x78
12
x3x5cx47x481–,x7cx7dx79xc (1.2.20)x18x18x11x12x13x14x60x1cx1dx7cx40x3
@M(x;y)
@y =
@N(x;y)
@x,(1:2:22)
x45x46x66,x40(1.2.20)x27x4cx74x75x76x33,x8x3ax1ax22x23 u(x;y) x51x6c(1.2.21)x72x4,
x5ex73x79 @u@x = M; @u@y = N,x57x55x11 M,N x2x6ex6fx4fx74x3bx73x6c My = uxy = Nx,
x28x7ex7f(1.2.22)x9(1.2.20)x27x4cx74x75x76x33x2x58x6fx13x6b.
x32x52,x7cx7dx3 @u@x = M x29x68x63x5cxx40x75(y x2x27x46x23x3ex6cu(x;y) x71x77xc
u(x;y) =
Z
M(x;y)dx+’(y); (1:2:23)
x58x49’(y)x27x6x2ax6ax36x2y x2x4fx74x22x23,x27x4ex36x30x3cx6a(1.2.23)x29x68x3y x59x2f
x73x23x3cx5fx59x7b @u@y = Nx3cx6c’(y)xdx37x38x2x76x33xc
@u
@y =
@
@y
Z
M(x;y)dx+ d’(y)dy = N(x;y);
x5ex73x6fx59’(y)x37x38
d’(y)
dy = N(x;y)?
@
@y
Z
M(x;y)dx,(1:2:24)
x60x67x56x6fx66x64x67x1x4e x x4bx63,x62x58x3 y x40x75x6x5x73x6c ’(y),x64x65(1.2.23)x24
x4cx4dx58x4cx74x75x27(1.2.20)x0x1x2 u(x;y),x16x45x66,x11x5c M(x;y) x6ex6fx4fx74,x27
x63xfx6dx13x6b(1.2.22)x7cx7dx79
@
@x
N(x;y)? @@y
Z
M(x;y)dx
= @N(x;y)@x? @@x
@
@y
Z
M(x;y)dx
= @N(x;y)@x? @@y
@
@x
Z
M(x;y)dx
= @N(x;y)@x? @M(x;y)@y · 0:
x57x55x62 (1.2.24) x29x68x3 y x40x75x6cx5a ’ (x1bx74x14x74x40x75x8x23) xdx64x65 (1.2.23)
x24x6cx4d(1.2.20)x2x27x9
u(x;y) =
Z
M(x;y)dx+’(y) = c (1:2:25)
x28x1bx43x68x1ex4dx47x48 2–,x55x30x6x5ax4dx13x6b(1.2.22)x2x1fx75x3bxdx7f.
x18x4cxbx39x31x27x9x65x4fx17x72xbx27
u(x;y) =
Z
N(x;y)dy +?(x) = c (1:2:26)
13
x58x49?(x)x11 @u@x = M x1ax4fx2cx20x6x2ax14x74x40x75x8x23x2x75x76x77x4ex36.
x76 1.2.8 x59x5ax76x33 (3x2y +8xy2)dx+(x3 +8x2y +12y2)dy = 0 x2x27
x9.
x2ax73x4 x28x29 M = 3x2y + 8xy2;N = x3 + 8x2y + 12y2,x11x5c @M@y =
3x2 +16xy = @N@x,x73x37x38x13x6b(1.2.22),x5cx9x11(1.2.26)x4fx6cx27x9
u(x;y) =
Z
N(x;y)dy = x3y +4x2y2 +4y3 +?(x):
x27x4ex36?(x),x62x66x64x29x68x3xx59x2fx73,x5fx59x7b @u@x = M,x4fx50x4fx16?(x) = 0x9
x24x17x27x9x9x3y +4x2y2 +4y = c.
x2ax73x17 x62x2bx76x33x75xex27
￡(3x2y +8xy2)dx+(x3 +8x2y)dy?+12y2dy = 0;
x8x3cx6cx6dx24x22x72x74x75xc
d(x3y +4x2y2)+4d(y3) = 0:
x5cx9x4fx4x6cx40x75.
x3y +4x2y2 +4y3 = c:
x28x69x75xdxex0x7x1ax59x4cx74x75x76x33x2x9x29x6ax6ax1ax77x6dx6fx3cx23x69x78x4fx17x7bx3bx59x1b
x9x4cx74x75x2x6xax76x33x2x9,x17x77x7cx7dx9x3fx5x51x7bx75xdxex0x7x3bx59x9x6xax74x75x76
x33.
2,x79x12x5bx19 x27x4dx68x1ex66x6cx2x47x48 3–,x7cx7dx4ex6fxexfx6x77x40x75x27x7dx2x3
x4,x51x52 (1.2.20) x1bx9x4cx74x75x76x33,x71x13x3ax1ax6ex6fx4fx74x22x23? =?(x;y) 6= 0,
x51x6c
(x;y)M(x;y)dx+?(x;y)N(x;y)dy = 0 (1:2:27)
x27x4cx74x75x76x33,x5ex73x3ax1ax22x23 u = u(x;y) x51x6c
du(x;y) =?(x;y)M(x;y)dx+?(x;y)N(x;y)dy;
x8x60?(x;y) x27x76x33(1.2.20)x2x79x12x5bx19,x28x29 u(x;y) = c x27(1.2.27)x2x27
x9,x65x9x76x33 (1.2.20) x2x27x9,x73x74,x79x29x51x40x75x27x7dx2x2bx27x4bx6fx23x2x22x23x65
x4fx50x9x76x33x2x44x9(x28x29x2bx76x33x24x17x28x40x75x27x7dx2cx6dx5cx2bx74x75x76x33x29x68x57x6bx17
x2x55x23x4ax6x2ax9)x28x2ax44x9x1bx6x36x3ax7ex1ax27x9x49.
x7cx71x3x5cx76x33 xdx + ydy + (x2 + y2)xdx = 0,x40x29x68x57x24?(x;y) =
1=(x2 +y2),x73x6c
xdx+ydy
x2 +y2 +xdx = 0;
x57x55
d(12 ln(x2 +y2))+d(12x2) = 0;
14
x11x63x73x7bx2bx76x33x2x27x9x27 x2 +ln(x2 +y2) = c.
xfx6dx13x6b(1.2.22),x22x23?(x;y) x27(1.2.20)x40x75x27x7dx2x1fx6fx13x6bx9
@(?M)
@y =
@(?N)
@x ;
x5ex73? =?(x;y) xdx37x38x71x77x2x6xax3ax3bx2fx74x75x76x33
N(x;y)@?@x?M(x;y)@?@y =
@M(x;y)
@y?
@N(x;y)
@x
,(1:2:28)
x1ax6x64x75x76x77,x6fx3ax59x7bx9(1.2.28)x3bx59x5a?,x57x55x6cx4d(1.2.20)x2x27x9,
x62x1ax59x9 (1.2.20) x10x25x2fx4f,x3dx9x1ax6x72x44x45x75x76x77,x78x9x1bx4fx59x5a (1.2.28)
x2x6x2ax44x9.
1– x71x13x3ax1ax56x4e x x79x63x2x40x75x27x7d? =?(x),x8x79 @?@y = 0,x5cx9
(1.2.28)x72x27N(x;y)d?dx = [My(x;y)?Nx(x;y)]?,x5ex73
d?
=
My(x;y)?Nx(x;y)
N(x;y) dx,(1:2:29)
x11x63x73x50,x27x4dx51x6cx76x33 (1.2.20) x3ax1ax56x4e x x79x63x2x40x75x27x7d? =?(x) x6d
x30x43x6d(1.2.29)x67x1 dx x2x2fx23x43x4e x x79x63,x73
My?Nx
N =,’(x),(1:2:30)
x71x13x28x2ax13x6bx72x4,x73x6c(1.2.20)x2x6x2ax56x4e x x79x63x2x40x75x27x7d
(x) = exp
Z
’(x)dx
,(1:2:31)
2– x57x39,x27x51 (1.2.20) x3ax1ax56x4e y x79x63x2x40x75x27x7d? =?(y) x6dx30x43x6d
x72x3ex64
Nx?My
M =,’(y)
x43x4e y x79x63,(x73x74x4dx66x72x3ex64x4e(1.2.30)x2x3x60x3bx4fx79x5ax5cx59x45)x71x13x28x2a
x13x6bx72x4,x73x6c?(y) = exp?R ’(y)dy￠.
xbx39x31,x6cx6dx53x54x57 (1.2.28) x70x5a,x27x51 (1.2.20) x3ax1ax56x4e x2 + y2 x79x63
x2x40x75x27x7d? =?(x2 +y2) x6dx30x43x6d My?NxyM?xN x56x4e x2 +y2 x79x63.
x4fx17xdx7f,x56x6fx76x33(1.2.20)x79x9,x24x58x36x3ax1ax40x75x27x7d?(x;y),x1fx2cxdx2
x40x75u(x;y) = c,x55x30x3x5cx14x15x6x2ax1bx8x5cx2x2x6ex6fx22x23F(u)x3c?(x;y)F(u(x;y))
x65x9x40x75x27x7d,x3xdx2x40x75x27 R F(u)du = c,x7cx71x76x33 ydx? xdy = 0,x79
x40x75x27x7d? = x?2,x6cx4dx40x75 u(x;y) =?x?1y = c x5cx9?(x)(?u?2) =
y?2,?(x)(?u)?1 = (xy)?1?(x)(1 § u2)?1 = (x2 § y2)?1 x8x17x9x40x75
15
x27x7d,x57x55x4fx17x75x9x59x5ax40x75,x=y = c,lnjx=yj = c,arctan(x=y) = c,
lnj(x?y)=(x+y)j = c.
x76 1.2.9 x59x9x76x33 (x4 +y4)dx?xy3dy = 0.
x2ax73x4 x60x67x79 M = x4+y4;N =?xy3;x11x63x50x5a My?Nx = 4y3+y3 =
5y3;(My?Nx)=N =?5=x,x73(1.2.30)x72x4,x12x11(1.2.31)x6c
(x) = exp
Z
5x?1dx
= x?5;
x62x63x40x75x27x7dx24x2bx76x33x29x68x6cx4cx74x75x76x33 (x?1 + x?5y4)dx?x?4y3dy = 0,
x5cx9x11(1.2.25)x73x6cx27x9ln(cx)?y4x?4=4 = 0,x73x74x51x40x75x27x7dx27x4bx6fx23x29
x2x = 0x65x9x9,x30x1bx3ax7ex1ax27x9x49.
x2ax73x17 x 6= 0x29,x7bx75xdxex0x9x7,x76x33x4fx7fx8x20x36x27
x4dx+y3(ydx?xdy) = x4dx?x2y3d(y=x) = x4dx?x5(y=x)3d(y=x) = 0:
x4fx50x4fx16x?5 x27x40x75x27x7d,x40x75x24x6cx27x9.
x2ax73x64 x27x76x33x9x6x2ax6ax5x76x33,x66x67x4fx17x18x6ax5x76x33 (1.2.6) x2x76x7x35,
x3dx9x65x4fx7bx40x75x27x7dx7,x4fx17xdx7f,x40 M(x;y)dx + N(x;y)dy = 0 x9x6x2ax6a
x5x76x33,x6d?,= (xM(x;y) + yN(x;y))?1 6= 0 x29,x8? x9x6x2ax40x75x27x7d,x40
· c 6= 0,x8x28x2ax6ax5x76x33x9x2ax4cx74x75x76x33.
x76 1.2.10 x59x9x76x33ydx+(y?x)dy = 0.
x2ax73x4 x11x5c M = y;N = y?x,x12x79 My?Nx = 2,x57x55 My?Nx?M =
2y x56x4e y x79x63,x5cx9x40x75x27x7d?(y) = exp(?2
Z
y?1dy) = y?2; x62x63x24
x2bx76x33x29x68x6cx4cx74x75x76x33 1ydx + (1y? xy2)dy = 0,x5cx9x11 (1.2.26) x6cx27x9
x
y + ln(cy) = 0,x73x74x51x40x75x27x7dx27x4bx6fx23x29x2 y = 0 x65x9x9,x30x1bx3ax7ex1ax27
x9x49.
x2ax73x17x75xdxex0x9x7,y 6= 0x29,x62x76x33x75xex27(ydx?xdy)+ydy = 0,x2b
x55x71x72
y2d
x
y
+ydy = 0;
x57x55x6dx6cx40x75x27x7dxcx40x75.
x2ax73x64 x27x76x33x9x6x2ax6ax5x76x33,(xM +yN)?1 = y?2 x9x40x75x27x7d,x57x55
x6dx6cx40x75.
x17x77x9x6x2ax40x75x27x7dx1bx56x9x13x5ex5cx6x2ax20x26x2x7cx7d,x7bx75xdxex0x7x77x6dx3e.
x76 1.2.11 x59x9x76x33 y0 =?x=y +
p
1+(x=y)2; (y > 0).
x2a x62x76x33x5ex71x72
xdx+ydy =
p
x2 +y2dx;
x69x1d
1
2d(x
2 +y2) = px2 +y2dx;
16
x11x63x73x50,x28x2ax76x33x79x40x75x27x7d? = 1px2 +y2,x17x63x24x76x33x29x68x6c
d(x2 +y2)
2px2 +y2 = dx;
x29x68x40x75x4fx6cx27x9 px2 +y2 = x+c,x1bx29x68x5cx76x21x6dx27 y2 = c(c+2x),xe
xdx6cx60x64x9y = (c(c+2x))1=2.
x6bx42x26x13x14x60x27x23x1dx13x2a
x29x6cxexfx2x6xax76x33,x17x9x4fx17x72xbx72x33x30x36x64x76x33 y0 = f(x;y)x2x6x72
x76x33,x3dx9x40x57x6x64x36x64x2x2cx76x33 F(x;y;y0) = 0 x4bx7x62 y0 x9x5a,x69x1dx73x51
x9x5a,x58x72x3ex64x65x39x8x29x2ax29,x31x63x66x38x31x72x76x7,x24x1bx50xdx7b,x7cx71Lagrange
x76x33
a(y0)y +b(y0)x = c(y0);
x24x9x28x69x75x36,x58x49 ax42 bx42 c,x17x9x20x26x2x6ex6fx4fx74x22x23,x27x4dx59x9x28xbx76x33,
x7cx7dx6fx3fx65x46x23 t,x51x6cx2cx76x33 F(x;y;y0) = 0 x50x7bx46x23x72xbx72x17x77x29x69x8x42
x36x64x52x6
y0 = p(t); y = f(x;p(t)); (1:2:32a;b)
y0 = p(t); x = f(y;p(t)),(1:2:33a;b)
x58x49p(t)x9tx2x7ax7bx4fx74x22x23.
x3x5cx36x64(1.2.32),x6fx3ax4ax20x26y,x68x4x20x26xx1ftx52x13x2x63x2fx64x3cx4fx17x62
(1.2.32b)x2x29x68x59x74x75x3cx6c
dy = fx(x;p)dx+fp(x;p)p0(t)dt; (1:2:34)
x2dx11(1.2.32a)x6c
dy = p(t)dt; (1:2:35)
x11(1.2.34)x1f(1.2.35)x3ax4ady x6cx,tx2x76x33:
fx(x;p)dx+[fp(x;p)p0(t)?p(t)]dt = 0:
x59x9x63x76x33,x26x58x27x9x1fx44x9x75x9x27Φ(x;t;c) = 0x1fΨ(x;t) = 0,x8x2bx76
x33x2x17tx27x46x23x2x27x9x27
‰ Φ(x;t;c) = 0;
y = f(x;p(t));
x44x9x27 ‰
Ψ(x;t) = 0;
y = f(x;p(t)):
x76 1.2.12 x59x76x33 y = y02?xy0 +x2=2 x2x9.
17
x2a x62x2bx76x33x21x27x8x42x36x64,y0 = t,y = t2?xt+x2=2,x3x19x1ax64x59x74x75
x5fx59x7bdy = tdxx6c
dy = 2tdt?tdx?xdt?tdx+xdx = tdx
x15x39x6c
(2t?x)(dt?dx) = 0:
x11x63x73x6c t?x = c x1f 2t?x = 0 x28x28 c x27x14x74x8x23,x6ax28x29x2ax64x7dx75x9x4e,
y = t2?xt+x2=2x2ex4,x73x6cx46x23x36x64x2x27x9
‰ x = t?c;
y = t2?xt+x2=2;
x1fx46x23x36x64x2x44x9 ‰
x = 2t;
y = t2?xt+x2=2:
x3ax4ax46x23tx75x9x6cx2bx76x33x2x60x64x27x9 y = x
2
2 +cx+c
2 x17xcx60x64x44x9 y = x2=4.
x76 1.2.13 x59x5ax36x37x38(Clairaut)x76x33
y = xy0 +f(y0)
x2x9,x58x49f(￠)x9x58x20x67x2x7ax7bx2x6ex6fx4fx74x22x23.
x2a x62x2bx76x33x21x27x8x42x36x64,y0 = t,y = xt + f(t); x3x19x1ax64x59x74x75x5f
x59x7bdy = tdxx6ctdx+xdt+f0(t)dt = tdxx15x39x6c
[x+f0(t)]dt = 0:
x73t = c,c x27x14x74x8x23;x1fx+f0(t) = 0,x62x30x7dx75x9x4e y = xt+f(t) x2ex4,
x73x6cx36x37x38x76x33x2x27x9 y = cx+f(c)x17xcx46x23x36x64x2x44x9
‰ x =?f0(t);
y =?tf0(t)+f(t):
x6d f00 6= 0 x29x3cx11x2cx22x23x36x39x3cx57x66x19x6x64x4fx7b t x27 x x2x22x23x3cx59x27
t = w(x),x57x55x4fx6cx60x64x44x9xc y = xw(x)+f(w(x)).
x3x5cx36x64(1.2.33),x6fx3ax4ax20x26x,x68x4x20x26y x1ftx52x13x2x63x2fx64x3cx4fx17x62
(1.2.33b)x2x29x68x59x74x75x3cx5fx59x7bdy = p(t)dx,x6c
dy(= p(t)dx) = p(t)[fy(y;p(t))dy +fp(y;p)p0(t)dt];
x59x9x63x76x33,x52x58x27x9x1fx44x9x75x9x27`(y;t;c) = 0x1f?(y;t) = 0,x5cx9x2bx76x33
x2x17tx27x46x23x2x27x9x27 ‰
`(y;t;c) = 0;
x = f(y;p(t));
18
x46x23x36x64x44x9x27 ‰
(y;t) = 0;
x = f(y;p(t));
x76 1.2.14 x59x5ax76x33 y03?4xyy0 +8y2 = 0 x2x9.
x2a x57x2bx76x33x9x5a x,x5fx26y0 = t:
x = y
03 +8y2
4yy0 =
t3 +8y2
4yt,(1:2:36)
x3(1.2.36)x2x29x68x59x74x75x3cx5fx59x7bdy = tdx,x15x39x5fx22x74x75x6c
t3?4y2
4y2t2 (2ydt?tdy) =
t3?4y2
4t3 d
t2
y
y,(1:2:37)
x11x63x6cx4dx27x9t2 = cy,x62x63x4e (1.2.36) x2ex4x3ax4atx73x6cx2bx76x33x2x27x9x27 y =
c(xc)2,x28x28?c = c=4 x27x14x74x8x23,x62t3?4y2 = 0 x4e(1.2.36)x2ex4x3ax4at
x6cx4dx2bx76x33x2x6x2ax44x9 y = 427x3,x30x60x67x1bx3ax7ex1ax27x9x49.
x73x74x3cx3x5cx6x64x2x2cx76x33x3cx6fx62x30x7dx21x27 (1.2.32) x69 (1.2.33) x2x36x64
x1ax77x55x4f,x3dx9x79x29xbx44x45x2x2cx76x33,x24x25x23x2cx18x11x12x13x14 F(x;y0) = 0 x1f
F(y;y0) = 0x3cx79x29x4fx17x77x16x1x6dx2x20x7dx3cx21x27 (1.2.32) x69 (1.2.33) x2x44x45x36
x64,x17x77x12x7cx7ex7fx58x9x7.
x76 1.2.15 x59x5ax76x33 y2(y0?1) = (2?y0)2 x2x9.
x2a x26 2?y0 = yt,x64x65x76x33x6c y2(y0?1) = y2t2,x3ax4a y2 x28x24x62x2bx76
x33x21x27(1.2.32)x2x36x64,y0 = p(t) = 1 + t2;y = 1t?t; x14x58x49x19x1ax64x2x67x68x1b
x60x7ex)x5cx9 p(t)dx(= dy) =?(1+t2)t?2dt,x12x79 dx =?t?2dt,x = 1t +c,
x27x55x2bx76x33x2x46x23x36x64x27x9x27
x = 1t +c; y = 1t?t;
x3ax4ax46x23 t x6cx60x64x27x9 y = x?c?(x?c)?1.
x76 1.2.16 x59x5ax76x33 x3 +y03?3xy0 = 0 x2x9.
x2a x79x7ax3x76x33x2bx4bx46x23x21,x26 y0 = tx,x64x65x2bx76x33x6c
x3 +t3x3?3tx2 = 0;
x11x63x9x6c x = 3t=(1+t3);x62x63x64x65y0 = txx73x6cy0 = 3t2=(1+t3);x28x24x62x2bx76
x33x21x27 (1.2.33)x2x36x64,y0 = p(t) = 3t2=(1+t3),x = 3t=(1+t3); x14x58x49x19
x1ax64x2x67x68x1bx60x7ey,x5cx9 dy = p(t)dx = 3t
2
1+t3
3(1?2t3)
(1+t3)2 =
9(1?2t3)t2
(1+t3)3 dt,
x57x55
y =
Z 9(1?2t3)t2
(1+t3)3 dt =
Z 3(3?2(1+t3))d(1+t3)
(1+t3)3 =
3(1+4t3)
2(1+t3)2 +c:
x27x63x2bx76x33x2x46x23x36x64x27x9x27
x = 3t1+t3;y = 3(1+4t
3)
2(1+t3)2 +c:
19
x1.3 x21x22x28x29x2ax27
x1ax1.1x49x7cx7dx7ax7x6x5ax7x2bx47x48x2x36x7,x3x5cx6xax76x33x3bx7ex24x9,
dx
dt = f(t;x); x(t0) = x0,(1:3:1a;b)
x79x7ax7cx7dx1dx6x2ax7cx7d,x761.3.1,x6xax74x75x76x33x2x7x2bx47x48
dx
dt = f(x) = 2x
1=2; x(0) = 0:
x2x9x79x29x2a,x = t2; t? 0x1fx = 0; t 2 (?1;+1);
x27x63,x50x1ax7cx7dx6cx29x4ex6fx4x73x9x4dx2x47x48x9xcx7x2bx47x48x2x9x9x79x3ax1ax78x40
x3ax1ax9x79xax6x78x19x6,x40x9x1bx3ax1a,x69x3ax1ax3dx1bxax6,x8x7ex7fx24x68x4x2x0x1x2ax79
x32x31x32x33x45x46x32x33x2x1dx1ex20x21,x27x27x45x46x32x33x9x33x34x3ax1ax2,x30x58x9x6ax6ax9x2c
x6x2,x19x1a,x11x5cx6x64x2x39x3ax3bx76x33x4bx7x17x4ex59x9,x27x63x56x50x57x76x33x10x25x3bx26
x27x9x2x3bx43,x69x1dx59x58x71x69x55x39x9,x69x1dx66x6ex2bx4bx23x2bx9x66x50;x40x47x48x2x9x1bx3a
x1a,x66x38x3bx2x24x4bx57x35x4f,x73x51x9x3ax1ax3dx1bx2cx6,x23x2bx66x50x65x4fx5cx2bx4b,x27x27x1b
x7bx4ex6fx66x50x2x9x36x6x2ax9x78x19x65,x40x24x10x7x2bx47x48x3ax1ax2cx6x9,x30x9x2x3bx43x2
x10x2cx0x45x46x32x33,x28x24x27x45x46x32x33x2x39x10x75x71x1fx37x65x26x27x38x36x4dx39x10x2x39,x19
x6b,x1axdx7fx7x2bx47x48x9x2x3ax1ax2cx6x3bx57x29,x6x64x65x27x58x55x39x69x23x2bx59x9x47x4ex4c
x7x1fx3ax63.
x58x5,x6d (1.3.1) x2x9x3ax1ax29,x9x4ex7x2b (t0;x0 x79x63,x7cx7dx6ax9x71x72 x =
x(t;t0;x0)x3cx11x5cx3bx26x2x3cx20,x7x2b (t0;x0 x6ax6ax9x55x39x2,x27x63x47x48 (1.3.1) x2
x9x56x9x45x46x32x33x2x28x69x55x39,x69x1dx7e(1.3.1)x2x9x9x34x7x2b(t0;x0)x20x21x2;x30
x9x79x27(t0;x0)x2x6ex6fx22x23? x40x1bx6ex6f,x59x9x47x48(1.3.1)x2x74x7x24x1bx23,x1ax71x3c
x73x51 (t0;x0) x4ex45x46x23x2bx2cx20x3cx70,x3dx11x5cx1bx6ex6fx3b,x6d (t0;x0) x20x21x29,x9 x
x4ex45x46x32x33x4fx50x2cx20x3cx23,x27x63x7cx7dx58x3fx3dx60x3excxdx3x47x48 (1.3.1) x14x66x62x63
x13x6b,x3fx38x17x31xdx58x9x3x7x2bx1fx46x23x2x6ex6fx2cx13x3bx78
xexd,x78x4ex2bx6x74x6ex6fx7x2bx47x48(1.3.1)x2x9x3x7x2bx2x4fx74x3b,x17xcx9x2xe
x23x3ax1ax5dx13x8x47x48.
x10x3x3x7x2bx47x48x9x2x3ax1ax2cx6x3bx6x5ax40x1dxdx7f,x55x3x58x41x47x48x11x5cx42x43x1f
x29x13x2x63x2f,x56x6x5ax16xdx4ex6fx2xex10.
x6x42x42x4ax24x25x2ax60x69x7x1fx4x11
x7dx4c 1.3.1 (Picard) x3x5cx7x2bx47x48 (1.3.1),x71x13x22x23 f(t;x) x37x38x13
x6bxc 1– x1ax79x41x7fx6bx36x5dx30 R = f(t;x) flfljt?t0j? fi;jx?x0j? fl;fi > 0;fl >
0g x66x6ex6f; 2– x1a R x66x63x5c x x37x38x6cx6dx74x6e (Lipschitz) x7cx40,x73x3ax1ax8x23
L > 0,x51x6c 8(t;x1);(t;x2) 2 R,x19x79
jf(t;x2)?f(t;x1)j? Ljx2?x1j (1:3:2)
x73
L(x2?x1)2? (x2?x1)(f(t;x2)?f(t;x1))? L(x2?x1)2 (1:3:2)?
20
x72x4,x8x7x2bx47x48(1.3.1)x1ax7fx5dx13I,= [t0?h;t0+h] x66x3ax1ax2cx6x9 x = ’(t).
x58x49 h = min(fi;fl=M);M = max
(t;x)2R
(jf(t;x)j).
x6fx70
(1) x79x7ax6ax59x7x2bx47x48(1.3.1)x2x9x21x27x59x40x75x76x33
x(t) = x0 +
Z t
t0
f(s;x(s))ds;t 2 I (1:3:3)
x2x6ex6fx9x3cx17x77xdx7f,x7x2bx47x48 (1.3.1) x2x9x1fx40x75x76x33 (1.3.3)x2x9x1aI x66x9
x2cx57x2,x12x56x4ex24x40x75x76x33(1.3.3)xdx7fx58x9x2x3ax1ax2cx6x3b.
xdxcx52x = x(t)x9(1.3.1)x2x9x3cx62x30x64x65(1.3.1a)xdx3cx29x68x63x5ctx57t0
x40x75x4dt,x5fx59x7bx7x21x13x6b(1.3.1.b),x73x6c(1.3.3).
x32x52x3cx52x(t)x9(1.3.3)x2x6ex6fx9x3cx6dt = t0 x29x3cx24x37x38x13x6b(1.3.1b),x62
(1.3.3)x29x68x3tx59x73x3cx24x6cx76x33(1.3.1a),xdx45.
(2) x2(1.3.3)x2Picardx72x7ax73x74x5bx5cxcx26 ’0(t),= x0,
’1(t),= x0 +Rtt0 f(s;’0(s))ds;
’2(t),= x0 +Rtt0 f(s;’1(s))ds;
...
’n(t),= x0 +Rtt0 f(s;’n?1(s))ds;
...
(1:3:4)
x17x77x59x7bx23x41x75x22x7xdx7f,x22x23x50x21 ’0(t),’1(t),:::,’n(t),:::x1ax5dx13I x66
x17x79x36x7x42x6ex6fx30x6x5dx79x41xc j’j(t)?x0j? fl j = 0,1,￠￠￠,t 2 I.
xdxcx6dk = 0x29x60x67x17x66xex10x72x4,x52k? 0x29x17x66xex10x72x4,x8
’k+1(t),= x0 +
Z t
t0
f(s;’k(s))ds
x1ax5dx13I x66x79x36x7,x6ex6fx3cx30x6x5dx79x41xc
j’k+1(t)?x0j =
flfl
flfl
Z t
t0
f(s;’k(s))ds
flfl
flfl? Mh? fl
xdx45.
(3)x67xdxdx7fx22x23x50x21 f’n(t)g x1a I x66x6x5dx4ex4f.x27x63x7cx7dx6a’n(t)?x0
x71x72x6x2ax1cx23x2x4dx75x1f:
’n(t)?x0 =
nX
k=1
[’k(t)?’k?1(t)]; t 2 I; n = 1;2;:::; (1:3:5)
x27x63xfx6d Weierstrass x2x1ax77x61x9x7x3cx7cx7dx56x3fxdx7f (1.3.5) x49x2x27xdx2x19x3
x2bx1bx23x5cx6x2ax4ex4fx2x31xdx1cx23x3xdx2x27xdx73x4f.
21
x79x7ax59x7bx23x41x75x22x7x1f Lipschitz x13x6b(1.3.2)xdx7f(1.3.5)x49x2x27xdx37x38
x71x77x1bx8x64xc
j’k(t)?’k?1(t)j? M0Lk?1jt?t0jk=k!,k? 1; (1:3:6)
x58x49
M0 = max
t2[t0;t0+h]
(jf(t;x0)j); (1:3:7)
xdxcx6dk = 1x29x3c
j’1(t)?’0(t)j =
flfl
flfl
Z t
t0
f(s;’0(s))ds
flfl
flfl? M0jt?t0j:
(1.3.6)x72x4,x52x6dk? 1x29(1.3.6)x72x4x3cx8
j’k+1(t)?’k(t)j =
flfl
flfl
Z t
t0
f(s;’k(s))?f(s;’k?1(s))ds
flfl
flfl
L
flfl
flfl
Z t
t0
j’k(s)?’k?1(s)jds
flfl
flfl? M0Lk=k!
flfl
flfl
Z t
t0
(s?t0)kds
flfl
flfl
= M0Lkjt?t0jk+1=(k +1)!:
x18x72x75x22xdx7f.
x5cx9x11x5c
j’n(t)?x0j?
nX
k=1
M0Lk?1hk=k!? M0(eLh?1)=L; 8n? 0
x73x17M0Lk?1hk=k!x27x27xdx2x31xdx1cx23x4ex4f,x57x55x50x21f’n(t)gx1aI x66x6x5dx4e
x4f.x30x79
j’n(t)?x0j? min(b;M0(eLh?1)=L),n = 1;2:::,(1:3:8)
x73,x57(1.3.7),(1.3.8)x4fx50x3cx40x74x75x76x33x49f(t;x)x1ax74x4cx7fx5dx30R = f(t;x) flfl
t 2 [a;b];x 2 (?1;+1)g x66x6ex6f; x1a R x66x63x5c x x37x38Lipschitzx8x23x27L
x2Lipschitzx13x6b,x8’n(t)x2x36x7x30x4fx17x9x15x2ax7fx5dx13I = [a;b].
(4) x59 limn!1’n(t) = ’(t),x8 ’(t) x2x27x6x5dx4ex4fx2x6ex6fx22x23x2x8x6fx22x23
x1a[t0;t0 +h]x66x9x6ex6fx2x30x79
j’(t)?x0j? min(b;M0(eLh?1)=L):
x11f(t;x)x1ax79x41x7fx5dx30Rx66x2x6x5dx6ex6fx3bx42xc’n(t)x1ax79x41x7fx5dx13Ix66x2x6x5d
x4ex4fx3bx73x4fx70x5afn(t),= f(t;’n(t))x1aI x66x2x6x5dx4ex4fx3b,x30x79 limn!1fn(t) =
f(t;’(t)),x5cx9x1a (1.3.4) x49x4fx26x7dx59x8x6fx1fx40x75x2x5x50,x29x68x26 n ! 1 x73
x6c ’(t) = x0 +Rtt0 f(s;’(s))ds,x28x24xdx7fx4d x = ’(t) x27 (1.3.1) x2x9,x5fx60
’n(t)x27x7x2bx47x48(1.3.1)x2x16nx7ax74x78x2a.
22
(5) xdx7fx9x2x2cx6x3b,x24x9x6fxdx7fx14x74x29x2ax7x2bx47x48 (1.3.1) x2x9x1ax30x7d
x2x5fx57x3ax1ax5dx13 ˉI ‰ I;(t0 2 ˉI) x66x9x7fx8x2,x7cx7dx7bx32xdx7,x40 (1.3.1) x78x79
x6x2ax9 x2 6· ’(t) =,x1; t 2 ˉI,
x26 r(t) = jx2?x1j2,x8
r0(t) = 2(x2?x1)(dx2=dt?dx1=dt) = (x2?x1)(f(t;x2)?f(t;x1); (1:3:9)
x11 (1.3.2)* x6c?L(x2? x? 1)2? r0(t)? L(x2? x1)2x3cx73?2Lr(t)?
y0(t)? 2Lr(t),x70x5a exp(?2L(t? t0))(r0(t)? 2Lr(t))? 0,xc exp(2L(t?
t0))(r0(t)62Lr(t))? 0,x73 ddt[r(t)exp(?2L(t?t0)]? 0; t? t0,xc ddt[r(t)exp(2L(t?
t0)]? 0; t? t0,
x5fx73x74x4dr(t0) = 0,x11x5cx22x23r(t)exp(?2Ljt?t0j)x1at? t0 x29x9x3ex56x4
x14x2,x1at? t0 x29x9x3ex56x21x6ax2,x73x6c r(t)exp(?2Ljt?t0j)? 0,x3dx27r(t)? 0,
x12r(t) · 0,x2cx6x3bxdx45.
2– x71x13x56x6fx59 f(t;x) x1a R x66x6ex6f,x7cx71x7x2bx47x48 dxdt = 2px,x(0) = 0,
x24x79x4bx6fx3fx2ax9
x =
( 0; t? c;
(t?c)2; t > c? 0:
x65x42x2ax60xdxe
x7dx51 x52 x = ’(t) x9x7x2bx47x48(1.3.1)x36x7x1ax5dx13I x66x2x6x2ax9,x40x3ax1a
(1.3.1)x2x4ax6x2ax9 x =?’ (t),x30x1ax5dx13J x66x79x36x7,x30x37x38xc
1,J I,x3d J 6= I;
2,?’ (t) · ’(t),x6d t 2 I;x8x60x9 ’(t),t 2 (fi;fl) x9x78xdxex60,x5fx60
’ (t) x9 ’(t) x1a J x66x2x6x2axdxe,x32x52,x40x1bx3ax1ax66x38x13x6bx2x9?’ (t),x24x60
x = ’(t),t 2 I,x9 (1.3.1) x2x6x2ax13x7ax2a,x5fx60 I x9x7x2bx47x48 (1.3.1) x2xex23
x3ax1ax5dx13,x73x74x3cx3x5cx1bx57x2x7x2bx13x6bx3cxex23x3ax1ax5dx13x4fx50x9x1bx2cx57x2.
x11x5cPicardx36x39x9xfx4dx3bx36x39,x30x7dx56x10x36x9x1a jt?t0j? h x66x3ax1a,x28
x28 h = min(a;b=M),x60x67x6d M x3cx23x29,h x24x3cx70,x28x1ax23x3fx23x75x76x77x6ax6ax1b
x50x37x38x3x9x2bx4bx75x71x2x4ex6f;x28x24x8x35x4dx47x48xcx50x79x62x36x7x1ax6x2ax70x5dx13x66x2
x9x11x12x4dx2fx23x2x5dx13x66x4ax78x62x63x58x2x9x3fx1bx50x1bx42x0x69x42x67x11x12x78
x7cx7dx1bx14xdx7fx31x6x5ax17x77xex10xc
xdxex7dx4c 1.3.2 x51x52 f(t;x) x1ax58x36x7x30 D x66x6ex6f,x30x1a D x66x79x41:
jf(t;x)j? M < +1,x58x49x36x7x30 D x4fx50x3ax7ex30x2x68x41 @D,x5fx30 D x2x1e
x4d(x1ex4x2x18x0)?D x9x6x2ax5dx30(x73x6ex27x2x20x18),x28x29x3cx7x2bx47x48 (1.3.1) x3ax1a
x9(x1bx6x36x2cx6)x3cx52x77x9x27 x = ’(t)x3cx58x36x7x5dx13 I x2x0x67x1x4x75x9x27 a,b,
x8x6dx3c (a;’(a)) 2?D x29,x58x4fx42x0x11x12;x6d(b;’(b)) 2?D,x58x4fx42x67x11x12,x5fx30
x40x = ’(t); t 2 I x9x47x48(1.3.1)x52x15x1fx9x2x58x6fx13x6bx9xc
x6dt ! a+0 x4e t ! b?0 x29x4(t;’(t)),(t;’(t)x75x9x17x42Dx2x68x41@D.
x6dDx9x5dx30x29x3cx17x66x13x6bx65x9x = ’(t); t 2 I x27x47x48(1.3.1)x52x15x1fx9
x2x1fx75x13x6b,x58x49,x40 D x9x4bx41x18x29x3cx75x27x4bx6fx38x4x9 D x2x68x41x4x52x6x3cx5f
x30x16x3ea =?1,b = +1.
23
x7cx71x76x33 dxdt = f(x) = 2x1=2 x58x36x7x30x9 (t;x) x5cx6cx49x2x66x71x5cx6cx3cx3a
x7ex68x41 x = 0,x58x37x38x7x21x13x6b x(0) = 1 x2x9x27 x = (t + 1)2; t1x29;
x = 0; t1x29; x30x2x3ax1ax5dx13x27 (?1;+1) x27x63x28x2ax9x9x15x1fx9,x6d
tx17x5cx0x1x4?1x29x3cx9x2x40x75x53x3ax66x3xdx2x4x17x42f x2x36x7x30x2x4bx6fx38x4
(?1;0),x6dtx17x5cx67x1x4+1x29x3cx9x2x40x75x53x3ax66x3xdx2x4x17x42(+1;+1),
x66x67x28x2ax4bx6fx38x4x2ax79x6x2ax4ex36x2x76x42.
x1ax28x2ax7cx7dx49x3cx6fx44x9x73x74x66x67x6dt =?1x29x3c x(?1) = 0x1ax68x41x66x3cx3d
x9x28x2ax9x78x9x4fx17x42x0x11x12,x27x63x6df(t;x)x2x36x7x30Dx3ax7ex68x41@D x29x3cx11
x12x36x39x49x2x58x6fx13x6bx1bx6x36x9x1fx75x2,x55x6df(t;x)x2x36x7x30x9x6x5dx30(x6ex27x2
x20x18)x29x3cx11x12x36x39x49x2x58x6fx13x6bx65x9x1fx75x13x6b.
x7cx71x76x33 dxdt = f(x) = (x2?1)=2 x30x2x36x7x30x9x6x2ax5dx30(x4cx5cx6c)x3cx37
x38x7x21x13x6b x(ln2) = 3 x2x9x27 x = 1+e
t
1?et x30x2x3ax1ax5dx13x27 (0;+1) x6d t
x17x5cx0x67x1x4x29x3cx9x2x40x75x53x3ax66x3xdx2x4x75x9x17x42 f x2x36x7x30x2x4bx6fx38x4
(0;+1)x1f(1;?1),x27x63x30x9x15x1fx9.
x6bx42x2ax5ex42x4ax60xfx10x47x18x11 x51x52 f(t;x) x1a (t;x) x2x5dx30 G x66x6ex6fx30
x51x52x3x5cx14x74x7x21x47x48(1.3.1)x9x9x3ax1ax30x2cx6x29,x79x17x77x36x39:
x7dx4c 1.3.3 x1ax17x66x51x52x77,x40x7x2bx47x48 x0 = f(t;x),x(?t0) =?x0 x2x9
x = ’(t;?t0;?x0) x1ax79x41x7fx5dx13 [a;b] x66x79x36x7,x8x30x2x27(t;t0;x0)x2x22x23,x9
(t0;x0)x2x6ex6fx22x23.
x56x42x2ax5ex42x4ax60x78x11x11 x6dx7cx7dx26x27x7x2bx47x48(1.3.1)x2x9 x = ’(t;t0;x0)
x29,x71x13x11x5cx3bx26x63x2fx55x51 x0 x20x21x29x3cx7cx7dx3ax4dx9x28x29x2x9x = ’(t;t0;x0)
x63x5c t0 x69x0 x2x20x21x6 @’@t
0
,@’@x
0
.
x27x63,x7cx7dx6ax40x75x76x33(1.3.3)x71x72
’(t;t0;x0) = x0 +
Z t
t0
f(s;’(s;t0;x0))ds;
x51x52@f(t;x)=@x x1a G x66x6ex6f,x8x7cx7dx24x4fx17x36x64x31x6ax66x64x29x68x3 t0 x1f x0
x75x9x59x73x6c
@’
@t0 =?f(t0;x0)+
Z t
t0
@f
@x(s;’(s;t0;x0))
@’
@t0(s;t0;x0)ds
x1f
@’
@x0 = 1+
Z t
t0
@f
@x(s;’(s;t0;x0))
@’
@x0(s;t0;x0)ds;
x11x63x73x50,@’@t
0
(t;t0;x0) x1f @’@x
0
(t;t0;x0) x2x27 t x2x22x23x9x77x21x3ax3bx6ax5
x76x33x75x9x37x38x7x21x13x6b ·(t0) =?f(t0;x0) x1f ·(t0) = 1 x2x9.
d·
dt =
@f
@x(t;’(t;t0;x0))·; (1:3:10)
24
x7cx7dx79x36x39xc
x7dx4c 1.1.4 x40x22x23 f(t;x) x1f @f@x(t;x) x17x1ax5dx30 G x66x6ex6f,x8x47x48
(1.3.1)x3ax1ax2cx6x2x15x1fx9 x = ’(t;t0;x0),x30x2x27 (t;t0;x0) x2x22x23x1ax58x20x21
x30x1ex9x6ex6fx4fx74x2; x63x6,@’@t
0
(t;t0;x0),@’@x
0
(t;t0;x0) x2x27 t x2x22x23x9x76x33
(1.3.10)x75x9x37x38x7x21x13x6b ·(t0) =?f(t0;x0);·(t0) = 1 x2x9.
25
x41x44x43 x32x33x34x35x36x37x38
x10x78x2x26x2x9x4cx6cx1fx40x1dx31x34x10x3ax3bx76x33x9x2xex24x1fx30x2x3bx43x17xcx8x2fx23
x76x33x2x59x9x76x7,x28x72x31x9x9x4dx3ax3bx0x1x2x6ex6fx3bx3c,x2dx9x26x27x39x3ax3bx47x48x58
x1bx4fx6ax2x2x39.
x2.1 x2bx29
x6xax76x33xex2x77x26x36x64x27 dxdt = f(t;x),x55x3cx1ax34x10x2x9x58x49x6x69x44x45x2
x36x64:
dx
dt = A(t)x+f(t); t 2 I; (2:1:1)
x58x49A(t)x27n￡n xax6bx25x2bx22x23xc A(t) = (aij(t))n￡n,x30Ax2x3dx2ax67x54aij
x9x1ax79x41x7fx5dx13I x66x6ex6f,x55 x x1f f(t) x27 t x2 n x66x77x26x2bx22x23,x77x26x2x3d
x2ax75x26x1aI x66x6ex6f,x11x5cx1ax36x64x66x76x33xex71x72x6x2ax8x64,x27x63x17xdx4fx6dx60x77x26
x36x64x23x69x6bx25x36x64x2x76x33xex27x76x33.
x27x4dx6ax1.3x49x63x5cx7x2bx47x48(1.3.1)x9x2x3ax1ax2cx6x3bx36x39x8x2x10x39x10x2xe
x13x27x4dx76x33 (2.1.1) x2x7x2bx47x48x66x3b,x58x63x7bx47x48x9x58x3fx9x4dx77x26x2bx22x23x50x21
x2x4ex4fx3bx47x48,x28x24x4ex6fx57x36x7x77x26x1fx6bx25x2x58x23x20x21,x58x23x9x33x26x2x19x3x2b
x2x70xc.
x6x42x20x5dx7ax2ax2bx12x2cx42x2d.
1,x20x5dx46x2ax2bx60x56x23 x52Xx27x23x30x66x24x79nx66x21x42x26x18x2ex27x42x26x14x7xe
x72x2x3ax3bx35x13x2x27x23x2x19x3x2bx1ax77x26x66x2x70xcx3cx36x7x77x26x,= (x1;:::;xn)T 2
X x2x6x69x56x23(x7bx15)x27
jxj,= jx1j+jx2j+￠￠￠+jxnj,(2:1:2)
x8x4fx50x28x58x36x7x2x58x23x37x38x19x3x2bx2x65x2ax2x10x3bx43:
1 jxj? 0
2,jxj = 0,x6dx30x43x6dx = 0
3,jx+yj?jxj+jyj
x24x17x6bx67x7bx27x8x19x3x2bx2x2cx2dx3bx72xbx58x23.
x3x6bx25A,x36x7x6bx25x2x58x23
jAj = max
jxj=1
(jAxj); (2:1:3)
x11x36x7(2.1.3)x4fx50x79x3bx43
(a) jAxj?jAjjxj:
26
x1fx5x59x7bx3bx43(a)x3cx4fx6cxcx40A,B x9x29x2ax57xax2x6bx25x3cx8x79x3bx43
(b) jABj?jAjjBj:
xd:
jABxj?jAjjBxj?jAjjBjjxj;
x11x36x7(2.1.3),x24x6cx3bx43(b).
x13x5x59x6bx25Ax2nx2ax21x42x26x27vj,j = 1,:::,n,x8xfx6dx36x7(2.1.3)x4fx17
xdx7fAx2x58x23x8x5cx28x72x21x42x26x2x58x23x49x2xex23x2bxcx73
jAj = maxj (jvjj),(2:1:4)
x45x46x66xfx6dx36x7(2.1.3)x3c
A = max
jxj=1
(jAxj) = max
jxj=1
j
X
j
jvjxjj? max(
X
j
jvjjjxjj)? maxj(jvjj)
x4ax6x76x6cx3cx52maxj (jvjj) = jvkj,x8x26xj = –jk x73x26xx2x19k x2ax67x54x27 1,x58
x41x67x54x27x2x3cx8
A?jvkj = maxj (jvjj)
x42x0x24x6cx2x29x2ax1bx8x64x73x6cxd(2.1.4).
x28x58 (2.1.4) x4fx17x2x27x6bx25x58x23x2x8x42x36x7x3cx27x63x28x58x36x7x2x6bx25x58x23x37
x38x4ex77x26x58x23x2cx57x65x2ax3bx43:x52A,B x57x27nxax6bx25,x8x79
1?A 0
2,jA = 0,x6dx30x43x6dA = 0
3,jA+Bj?jAj+jBj
2,x20x5dx46x2ax2bx5bx5cx60x32x33x11 x3x5cx36x7x1ax79x41x7fx5dx13 I,= [a;b] x66x2
x24x79nx66x6ex6fx22x23x77x26xex72x2x3ax3bx35x13,x7cx7dx4fx17x36x7x30x2x58x23x27
kxk,= max
t2I
(jx(t)j);
x35x13x49x4x21x2x4ex4fx3bx45x46x66x24x9 n x66x22x23x77x26x50x21x18x75x26x2x6x5dx4ex4fx3b,n
xax6ex6fx22x23x6bx25x3ax3bx35x13x65x79x57x58x2xex13.
3,x49x23x20x5dx46x49x23x2ax2bx60x48x10x7ax79x12 x7cx7dx7ax7x7bx4ex6x2a n x34x49x23x20
x5dx60x10x23x24x9x3x30x2x3dx2ax75x26x59x73x6cx4dx2x22x23x77x26,x57x58,x49x23x2ax2bx60x10x23x24
x9x3x30x2x3dx2ax67x54x59x73x6cx4dx2x6bx25x22x23,xbx39x31,x49x23x20x5dx60x79x12x24x9x3x30x2
x3dx2ax75x26x2bx4bx40x75x6cx4dx2x77x26,x55x49x23x2ax2bx60x79x12,x24x9x3x30x2x3dx2ax67x54x2bx4b
x40x75x6cx4dx2x6bx25,x5cx9x3x5c n xax76x25 A(t) x1f B(t) x17xc n x66x77x26 u(t) x1f
v(t) x7cx7dx79
[A(t)+B(t)]0 = A0(t)+B0(t);
[u(t)+v(t)]0 = u0(t)+v0(t);
[A(t)B(t)]0 = A0(t)B(t)+A(t)B0(t);
27
[A(t)u(t)]0 = A0(t)u(t)+A(t)u0(t):
4,Lipschitz x7cx40 x40 n x66x22x23x77x26 f(t;x) x1ax79x41x7fx5dx30
R = f(t;x)jt 2 R;x 2 X;jt?t0j? a;jx?x0j? b;a > 0;b > 0g
x66x6ex6f,x30x3ax1ax8x23 L > 0,x51x6c 8(t;x1);(t;x2) 2R,x79
jf(t;x1)?f(t;x2)j? Ljx1?x2j;
x8x60 f(t;x) x1a Rx66x63x5c x x37x38x17 L x27Lipschitzx8x23x2Lipschitzx13x6b.
x63x6,xbx39x31x4fxdx7fx76x33xex2 Cauchy x47x48
x0 = f(t;x); x(t0) = x0 (2:1:5)
x2x9x4ex40x75x76x33
x(t) = x0 +
Z t
t0
f(s;x(s))ds
x2x6ex6fx9x9x2cx57x2.
x69x63,x1.3x49x63x5cx7x2bx47x48x9x2x3ax1ax2cx6x3bx42x9x2x11x12x17xcx9x3x7x2bx1fx46
x23x2x6ex6fx3bx1fx4fx74x3bx2xdx7fx32x33x1fxex13,x1bx35x4fx17x1bx2x14x15x36x5ex31x27x4dx7x2bx47
x48(2.1.5)x66x3b,x56x6fx62xx5ex27x77x26x,x22x23f(t;x)x5ex27x22x23x77x26f(t;x),x22
x23x2x19x3x2bx39x9x72x77x26x2x58x23x73x4f.
x1ax42x22x11x13x14x24x42x4ax24x25x2ax60x69x7x1fx4x11x7dx4c
x11x5cx3ax3bx76x33(2.1.1)x67x1x63x5cxx9x3ax3bx2,x40 n xax22x23x6bx25 A(t) x1f n
x66x22x23x77x26 f(x) x1ax79x41x7fx5dx13 [a;b] x66x6ex6f,x8 A(t)x+f(t) x25x67x37x38x17
L = max
t2[a;b]
jA(t)j x27Lipschitzx8x23x2Lipschitzx13x6b,x3ax3bx76x33(2.1.1)x2x67x1
x1a“x74x4c”x7fx5dx30 [a;b]￡X x66x6ex6fx3cx27x63x6ex6fx4d (1.3.8) xdx2x73x3cxbx39x5c x1.3
x49x59x7bPicardx1fx5x7cx55x7x2xdx7f,x4fx6cx4cxfx3bx2x3ax1ax4ex2cx6x3bx36x39:
x7dx4c 2.1.1 x40 n xax76x25 A(t) x1f n x66x77x26 f(x) x1ax5dx13 I,= [a;b] x66
x6ex6f,x8 8t0 2 I x1f 8x0 2 X,x7x2bx47x48
x0 = A(t)x+f(t);x(t0) = x0
x7x38x5x5x4aI x66x3ax1ax2cx6x9 x = ’(t).
x65x42x13x14x24x46x39x19x13x14
x3x5cx33x30x36x64x2x25xax76x33(1.1.6),
x(n) = f(t;x;x0;￠￠￠x(n?1)),(2:1:6)
28
x5cx9x40x26 x = x1;x01 = x2;￠￠￠ ;x0n?1 = xn,x8(2.1.6)x24x72x27x76x33xe
8
>>>>
><
>>>>
>:
x01 = x2;
x02 = x3;
...
x0n?1 = xn;
x0n = f(t;x1;x2;￠￠￠ ;xn);
(2:1:7)
x59 x = (x1;:::;xn)T,f(t;x) = (x2;x3;:::;xn;f(t;x))T,x8 (2.1.7) x24
x9(1.1.8),x73 x0 = f(t;x),x1bx4fx1dx5a,x40 x = ’(t) x9(2.1.7)x2x9x22x23x77x26,
x8x58x19x6x2ax75x26x24x9 (2.1.6) x2x9; x32x52,x71x13 ’(t)x9 (2.1.6) x2x9,x8 (’(t),
’0(t),:::,’(n?1)(t))T,x24x9 (2.1.7) x2x9x22x23x77x26,x28x24x9x25xax76x33x21x72x76
x33xex2x6x64x76x7x17xcx30x7dx9x52x13x2x63x2f,x32x32x3b,x6x64x76x33xex9x21x1bx72x25xax76
x33x2,x3dx9x65x79x6x72x7cx6,x7cx71x76x33xe x01 = x2,x02 = x1 x24x4fx21x72x1axax76x33
x001 = x1 x2bx4bx59x9.
x3x5c n xax3ax3bx76x33(1.1.9),x6d a0(x) 6= 0 x29,x7cx7dx4fx17x6ax30x71x72x33x30x36x64
x(n) +a1(t)x(n?1) +￠￠￠+an?1(t)x0 +an(t)x = f(t); (2:1:8)
x28x29x59x7bx66x6cx2x4cx7,x4fx62(2.1.8)x21x72x3ax3bx76x33xe(1.1.10),x73 x0 = A(t)x+
f(t),x58x49
A(t) =
2
66
66
64
0 1 0 ￠￠￠ 0
0 0 1 ￠￠￠ 0
...,..,..,..
0 0 0 ￠￠￠ 1
an(t)?an?1(t)?an?2(t) ￠￠￠?a1(t)
3
77
77
75; f(t) =
2
66
66
64
0
0
...
0
f(t)
3
77
77
75:
x55 n xax76x33(2.1.6)x69(2.1.8)x2 n x2ax7x21x13x6bx27
x(t0) = x01;x0(t0) = x02;￠￠￠ ;x(n?1)(t0) = x0n; (2:1:9)
x3x5cx76x33xe x0 = f(t;x) x69 x0 = A(t)x+f(t) x3bx7e,x4ex63x2cx3xdx2x7x21x13x6b
x24x9 x(t0) = x0,x28x28 x0 = (x01;x02;￠￠￠ ;x0n)T.
x3x5cx1axax3ax3ax3bx76x33
d2x
dt2 +f(x)
dx
dt +g(x) = 0; (2:1:10)
x78x7x8x2bx7bx24x25x2 Li′enard x5cx3bx62x58x21x72x76x33xe,x73x26 F(x) =
Z x
0
f(x)dx,
y = dxdt +F(x),x5cx9x76x33(2.1.10)x24x72x27x76x33xe
dx
dt = y?F(x);
dy
dt =?g(x);
x28x28 x;y x5dx27x33x26.
x19x52,x14x15x25xax76x33xcx58x36x9x47x48x17x4fx17x21x72x76x33xexcx58x36x9x47x48x2bx4bx59
x9,x3dx6x64x3bx7e,x32x52x1bx67.
29
x2.2x2cx2dx28x29
x7cx7dx3cx1ax68x4dx3ax3bx76x33 (2.1.1) x2x6x64x39x10,x6ex6fx9x26x27x30x2x9x2xex24x47
x48.x1a(2.1.1)x49,x71x13 f(t) 6· 0,x7cx7dx24x60(2.1.1)x27x22x11x23x79x7ax13x14,x60 f(t)
x27x23x79x7axb,x71x13 f(t) · 0;a? t? b,x8x76x33(2.1.1)x72x27
x0 = A(t)x; (2:2:1)
x28x29x60x30x27x22x11x79x7ax13x14;x55x30x27x8x6a(2.2.1)x60x27(2.1.1)x5ex45x60x22x11x79x7ax13
x14.
x6x42x22x11x79x7ax13x14x24
x10x1fx6ex6fx26x27 (2.2.1) x9x18x2x64x23xex24x47x48,x27x63x7cx7dx19x51x52 n xax76x25
A(t) x1ax79x41x7fx5dx13 I,= [a;b] x66x6ex6f.
x52 u(t) x1f v(t) x17x9 (2.2.1) x2x9,fi x1f fl x9x29x2ax14x74x8x23,x53x54x64x65
(2.2.1)x4dxdx73x7b,fiu(t)+flv(t) x65x9(2.2.1)x2x9,x28x24x9x3ax3bx6ax5x76x33x2x3e
x14x2bx39.
x7dx4c 2.2.1 (x3fx6x62x4c) x3ax3bx6ax5x76x33(2.2.1)x2x14x15x29x2ax9x2x3ax3bxex0
x78x9x30x2x9(x57x55x14x74x79x6fx2ax9x2x3ax3bxex0x78x9x9).
x28x13x36x39x7ex7f,(2.2.1) x2x9x18x24x72x6x2ax22x11x52x4a,x25x67x6fx47xc (2.2.1)x2x27
x9x9x79x4fx17x7bx79x6fx2ax44x9x2x3ax3bxex0x3bx72xbx3cx71x4fx17x3cx4ex6fx3fx6ax2ax28x69x44x9?
x27x63x7cx7dx4ex6fx3fx2bx22x23x77x26x2x3ax3bx2cx63x1fx3ax3bx4bx63x3x4,x30xbx39x5cx3ax3bx64x23x49
x77x26x2x3ax3bx2cx63x1fx3ax3bx4bx63x3x4x2x70xc,x24x1bx57x2x9x3cx3ax3bx64x23x49x2x34x10x2x77
x26x9x8x26x3cx55x3cx1ax34x10x2x9x20x21x2x77x26–x22x23x77x26,x27x63x36x7x66x79x24x1bx57.
x7cx7dx60x36x7x1ax5dx13 I x66x2 m x2a n x66x22x23x77x26 x1(t),x2(t),￠￠￠,xm(t)
x9x22x11x47x77x2,x71x13x3ax1a m x2ax1bx4cx27x2x2x8x23 c1,c2,￠￠￠,cm x51x6cx1aI x66,
x72x4x7fx8x64
c1x1(t)+c2x2(t)+￠￠￠+cmxm(t) · 0; (2:2:2);x79x8,x24x60 x1(t),x2(t),￠￠￠,xm(t) x1a I x66x22x11x40x77.
x73x74xfx6dx3ax3bx64x23x49x6bx25x1fx21x77x26x2cx24x2x7x8x3c (2.2.2) x65x4fx17x6dx71x72x6b
x25x1fx77x26x2cx24x2x36x64xc
Xc · 0; (2:2:3)
x58x49x6bx25X = (x1;x2;:::;xm)x27m x2a n x66x22x23x77x26x1(t),x2(t),￠￠￠,
xm(t)x13x5x43x72x2x6bx25,x77x26 c(c1;c2;:::;cm)T x27 m x66x8x23x21x77x26,x28x29x66
x38x3ax3bx2cx63x3bx4fx17x37x38x27xcx76x33(2.2.3)x1ax15x2ax5dx13I x66x3ax1ax39x2x8x77x26x9c.
x7cx71x22x23x77x26
(cos2 t;0;￠￠￠ ;0)T x1f(sin2 t?1;0;￠￠￠ ;0)T
x4fx16c = (1;1)T 6= 0,x27x63x1a t x2x14x15x5dx13x66x17x9x3ax3bx2cx63x2.
30
x6d n x2a n x66x22x23x77x26x1ax5dx13 I x66x3ax3bx2cx63x29x3cx11 (2.2.3),xfx6dx3ax3bx64x23
x7bx29,x58x79 detX · 0; t 2 I,x7cx7dx60x6bx25 X x2x4bx21x64x27x28 n x2ax22x23x77x26x2
x41x42x2(Wronsky)x4bx21x64,x59x2W(t):
W(t),= detX:
x11x3ax3bx64x23x7bx29x7cx7dx7bx4ex3cx40x6x2ax76x25x2x4bx21x64x8x5cx2x3cx8x76x25x2x21x42x26
x3ax3bx2cx63,x3dx9x28x3x22x23x77x26x5ex7ex1bx6x36x3x3cx73x77x26x2 Wronsky x4bx21x64x2x2b
x1a I x66x7fx8x5cx2x3cx5fx1bx50x70x5ax77x26x2x3ax3bx2cx63x3bx3cx2bx27x24x1ax5cx22x23x77x26x6x64
x9x20x26,x1bx6x36x3x24x79x2x20x26t (2.2.3)x3ax1ax6x2ax8x23x77x26x9x3c
x7cx71,x3x14x15x15x23 n > 0,x77x6cx2 n x2a n x66x22x23x77x26
2
66
66
64
1
0
0
...
0
3
77
77
75;
2
66
66
64
t
0
0
...
0
3
77
77
75;
2
66
66
64
t2
0
0
...
0
3
77
77
75;￠￠￠ ;
2
66
66
64
tn?1
0
0
...
0
3
77
77
75
x1a t x2x14x15x5dx13x66x17x9x3ax3bx4bx63x2,x66x67x41x7dx2Wronskyx4bx21x64x2x2bx7fx27x2.
x67x55,x40nx2anx66x22x23x77x26x1(t),x2(t),:::,xn(t)x9x3ax3bx76x33(2.2.1)x2
x9x77x26x29,x24x79
x7dx4c 2.2.2 x71x13(2.2.1)x2x9x77x26 x1(t),x2(t),￠￠￠,xn(t) x2Wronsky
x4bx21x64x1aI x49x6x4t0 x2x2bx8x5cx2xc (x73W(t0) = 0),x8x1(t),x2(t),:::,xn(t)
x1aI x66x3ax3bx2cx63x14x57x55Wronskyx4bx21x64x1aI x66x7fx27x2).
x6fx70 x11x36x39x51x52x3c W(t0) = 0,x8x76x33xc
X(t0)c = 0 (2:2:4)
x2x2fx23x4bx21x64x2x2bx27x2,(2.2.4)x3ax1ax39x2x21x77x26x9c;x17x63x24x25x22x23x77x26
x(t),= X(t)c:
x11x3ex14x2bx39x7b x(t) x27 (2.2.1) x2x9,x30x11 (2.2.4) x4fx7bx30x78x37x38x7x21x13x6b
x(t0) = 0,x55x6dx36x39 2.1.1 x7bx4e,(2.2.1) x37x38x2x7x21x13x6bx2x2cx6x9x27x2x9,
x73 x(t) · 0 x3 t 2 I,x5ex73x3ax1ax39x2x2x21x77x26c,x51X(t)c · 0;t 2 I x73x1(t),
x2(t),￠￠￠,xn(t)x1aI x66x9x3ax3bx2cx63x2,x36x39xdx45.
x57x36x39 (2.2.2)x2xdx7fx4fx50x3c (2.2.1) x2x9x77x26 x1(t);x2(t);￠￠￠ ;xn(t) x2
Wronsky x4bx21x64x1ax15x2ax5dx13x66x6fx63x7fx8x5cx2,x6fx63x7fx1bx8x5cx2x9x27x63x7cx7dx6c
x4dx6x2ax61x1c (2.2.1) x2 n x2ax9x9x3ax3bx2cx63x78x9x3ax3bx4bx63x2x4cx7,x14x74x16x6x4
t0 2 I,x40x30x7dx2 Wronsky x4bx21x64x1ax28x4x2x2b W(t0) = 0,x8x28 n x2ax3ax3bx2c
x63;x40W(t0) 6= 0,x8x28 n x2ax9x3ax3bx4bx63.
x7dx4c 2.2.3 x3ax3bx76x33(2.2.1)x58x36x3ax1a n x2ax3ax3bx4bx63x9 x1(t),￠￠￠,xn(t).
31
x6fx70 x14x16 t0 2 [a;b],x5cx9x11x36x39(2.2.1)x73x7b(2.2.1)x75x9x37x38x7x21x13x6b
x1(t0) =
2
66
66
64
1
0
0
...
0
3
77
77
75;x2(t) =
2
66
66
64
0
1
0
...
0
3
77
77
75;￠￠￠ ;xn(t0) =
2
66
66
64
0
0
...
0
1
3
77
77
75;
x2x9 x1(t),￠￠￠,xn(t) x58x36x75x9x1a [a;b] x66x3ax1ax2cx6,x2dx27x27x28 n x2ax9
x2 Wronsky x4bx21x64 W(t) x1a t = t0 x5x37x38 W(t0) = 1 6= 0,x27x63x1a
[a;b]x66W(t) 6= 0,x57x55x28 n x2ax9x3ax3bx4bx63,x36x39xdx45.
x7dx4c 2.2.4 x71x13 x1(t);x2(t);￠￠￠ ;xn(t) x9(2.2.1)x2 n x2ax3ax3bx4bx63x9,
x8(2.2.1)x2x14x6x9 x(t) x17x4fx17x7bx28 n x2ax9x2x3ax3bxex0x3bx72xb
x(t) = X(t)c = (x1(t);x2(t);:::;xn(t))c
x58x49 c = (c1;c2;￠￠￠ ;cn)T x9x8x23x77x26.
x6fx70 x14x16 t0 2 [a;b],x6ex6fx17 c = (c1;￠￠￠ ;cn)T x27x7fx7bx77x26x2x76x33
X(t0)c = x(t0); (2:2:5)
x58x2fx23x4bx21x64x31x3dx9 W(t0) 6= 0,x27x63(2.2.5)x79x2cx6x39x2x9 c,x11x63x36x7x22x23
x77x26 ’(t),= X(t)c,x11x3ex14x2bx39 ’(t) x65x9(2.2.1)x2x9,x5cx9x11(2.2.5)x73x7b
x(t0) = ’(t0),x57x55x11x36x392.1.1x2x2cx6x3bx70x5a
x(t) · ’(t) = X(t)c,(2?199)
x36x39xdx45.
x36x1d x3ax3bx76x33(2.2.1)x2x3ax3bx4bx63x9x2xex23x2ax23x27n,x30x76x33(2.2.1)x2x27
x9x11X(t)cx72xb,x30x3ax7ex4dx6x45x9.
x11x36x392.2.1x1fx36x392.2.4x73x7b,x13x14(2.2.1)x35x2ax2ax60x67x47x48x51x4x5nx34
x60x22x11x52x4a.x7cx7dx60x76x33(2.2.1)x2x14x15nx2ax3ax3bx4bx63x9x27x30x2x6x2ax7fx52x2ax24;
x60x67x3ax1ax4dx4bx6fx3fx2ax1bx57x2x2x10x9xe.
x17x9x7fx52x2ax2ax2b
x62x76x33 (2.2.1) x2 n x2ax9x77x26 ’1(t),’2(t),￠￠￠,’n(t) x2x27x21x77x26xex72
x2 n xax76x25 Φ(t) = [’1(t);’2(t);:::;’n(t)] x60x27(2.2.1)x2x2ax2ax2b,x71x13x28
n x2ax9x77x26x78x3ax3bx4bx63,x73x24x72x76x33(2.2.1)x2x6x2ax2x10x9xe,x31x63x28x29x2x9x6b
x25Φ(t),x24x60x27x76x33(2.2.1)x2x7fx52x2ax2ax2b,x69x6dx60x27x7fx2ax2b,x44x9x6dΦ(0) = I
x29,x60 Φ(t) x27(2.2.1)x2x1ex7dx7fx2ax2b.
x4fx50(2.2.1)x2x9x6bx25x37x38x17x77x6bx25x74x75x76x33:
X0 = AX; (2:2:6)
x3x5cx28x2ax6bx25x74x75x76x33x2x7x2bx47x48x2x9x65x79x2cxdx2x4cxfx3ax1ax1fx2cx6x3bx2x36x39.
32
x7dx4c 2.2.5 x40 Φ(t) x27x76x33(2.2.1)x2x2x9x25,x8x3x14x74x2x39x1fx4fx8x76x25
C (x73detC 6= 0),Φ(t)C x65x9(2.2.1)x2x2x9x25.
xdxcx26X = ΦC,x64x65(2.2.6)x64x6c:
X0 = (ΦC)0 = Φ0C = A(ΦC) = AX
x28x7ex7fx9x6bx25x24x17x8x23x6bx25x78x9x9x6bx25x3cx6dx9x6bx25x78x9x2x9x25x29x3cx11x5c X x1f
C x17x9x39x1fx4fx2x6bx25x3cx30x7dx2x24x40x65x9x39x1fx4fx2,x24x17x9x2x9x25,xdx45.
x73x74x6dx6bx25x2xax23x23x5cx6x29,x8x23x6bx25x24x17x9x6bx25x1bx6x36x9x9x6bx25.
x7dx4c 2.2.6 x40 Φ(t) x1f Ψ(t) x5dx27(2.2.1)x2x2x9x25,x8x58x3ax1ax39x4fx2 n
xax8x76x25 C,x51x6c Ψ(t) = Φ(t)C,x3 t 2 I.
x6fx70 x11x5cx2x9x25 Φ(t) x3t0 2 I x39x4f,x8 Φ?1(t0) x3ax1ax5ex39x4f,x16x8x23
x39x1fx4fx25
C,= Φ?1(t0)Ψ(t0)
xfx6dx36x392.2.5,Γ(t) = Φ(t)C x65x9x2x9x25,x5fx30x1at = t0 x29x4ex9x6bx25Ψ(t)x37
x38x2cx57x2x7x21x13x6b:
Ψ(t0) = Γ(t0)
x11x9x2x2cx6x3bx3c
Ψ(t) · Γ(t) = Φ(t)C
xdx45.
x76 2.2.1 x4dxd Φ(t) =
et tet
0 et
x27x76x33 x0 =
1 1
0 1
x x2x2x9x25.
x6fx70 x79x7ax62 Φ(t) x53x54x3 t x59x73x4fx6c Φ0(t) =
et (1+t)et
0 et
,x67xd
x50x5aAΦ(t) =
1 1
0 1
et tet
0 et
=
et (1+t)et
0 et
,x11x63x73x50 Φ(t) x27x6b
x25x76x33(2.2.6)x2x9(x76x25),x57x55x9(2.2.1)x2x9x6bx25,x63x6,detΦ(0) = 1 6= 0,
x24x17 Φ(t) x27(2.2.1)x2x2x9x25.
x65x42x22x11x23x79x7ax13x14x24
x7cx7dx3cx1ax68x4dx3ax3bx39x6ax5x76x33(2.1.1),x73 x0 = A(t)x+f(t) x9x2xex24x47
x48.x28x28 n xax2fx23x22x23x6bx25 A(t) x1fx39x6ax5xd n x66x22x23x77x26 f(t) x17x51x52x1a
x5dx13 I,= [a;b] x66x6ex6f,x53x54x4dxdx4fx6cx77x21x29x2ax50x48xc
x51x25 2.2.1 x40 ’(t) x27x39x6ax5x76x33(2.1.1)x2x9,?(t) x27(2.1.1)x24x3xd
x2x6ax5x76x33(2.2.1)x2x9,x8 ’(t)+?(t) x27x39x6ax5x76x33(2.1.1)x2x9.
x51x25 2.2.2 x40 ’(t) x1f?(t) x17x9x39x6ax5x76x33(2.1.1)x2x9,x8x58x20 ’(t)?
(t) x27x58x3xdx2x6ax5x76x33(2.2.1)x2x9.
x7dx4c 2.2.7 x52 Φ(t) x27x6ax5x76x33 (2.2.1) x2x2x9x25,’(t) x27x39x6ax5x76x33
(2.1.1)x2x14x6x44x9,x8x39x6ax5x76x33(2.1.1)x2x27x9x27
x = Φ(t)c+’(t); (2:2:7)
33
x58x49 c x27x14x74 n x66x8x77x26;x30(2.2.7)x3ax7ex4d(2.1.1)x2x6x45x9.
x6fx70 x79x7ax11x50x482.2.1x4fx7b,(2.2.7)x27(2.1.1)x2x9;x58x5x11x5c Φ(t) x2x39
x1fx4fx3b,x5cx9x3x5c(2.1.1)x2x14x6x9?(t),x16(2.2.7)x49x2 c = Φ?1(t0)(?(t0)?
’(t0));t0 2 [a;b],x8 Φ(t)Φ?1(t0)(?(t0)?’(t0))+’(t) x1f?(t) x5dx27(2.1.1)
x37x38 x(t0) =?(t0) x2x9,x27x63x11x36x39 2.1.1 x49x9x2x2cx6x3bx73x7bx3 t 2 [a;b]
x79
(t) · Φ(t)Φ?1(t0)(?(t0)?’(t0))+’(t):
x36x39xdx45.
x11x63x36x39x73x50,x27x59 (2.1.1) x2x27x9,x79x7ax59x5ax58x3xdx6ax5x76x33 (2.2.1) x2
x6x2ax2x9x25,x1bx7bx8x23x20x6dx7x59x5ax6x2ax44:.
x52 Φ(t) x27 (2.1.1) x24x3xdx2x6ax5x76x33 (2.2.1) x2x6x2ax2x9x25,x7cx7dx3bx59
(2.1.1)x36x71
’(t) = Φ(t)c(t) (2:2:8)
x2x9,x58x49 c(t) x27x6ax36x2 n x66x22x23x77x26;x62(2.2.8)x64x65(2.1.1)x73x6c
Φ0(t)c(t)+Φ(t)c0(t) = A(t)Φ(t)c(t)+f(t);
x11x5c Φ0(t) = A(t)Φ(t),x12x6c
c0(t) = Φ?1(t)f(t);
x29x68x40x75x6c
c(t) = c0 +
Z t
t0
Φ?1(s)f(s)ds;
x28x28 c0 x27x14x74x8x77x26,x62x63x64x65(2.2.8)x6cx4d(2.1.1)x2x27x9x27 (x32x23x5cxax52
x13)
x = Φ(t)
c0 +
Z t
t0
Φ?1(s)f(s)ds
,(2:2:9)
x3x5cx6x36x2 t0 2 [a;b],x8(2.1.1)x37x38 x(t0) = x0 x2x9x27
’(t) = Φ(t)Φ?1(t0)x0 +
Z t
t0
Φ(t)Φ?1(s)f(s)ds,(2:2:10)
x76 2.2.2 x59x5ax7x2bx47x48x0 = Ax+e?tv1,x(0) = v2 x2x9x3cx58x49
A
1 1
0 1; v1 =
1
0; v2 =
1
1
:
x2x9.
x2ax73x4 x10x48x49x76x33x24x3xdx2x6ax5x76x33x27x7c2.2.1x49x2x76x33,x11x7c2.2.1x7b
x4e,x58x2x9x25x27
Φ(t) =
1 t
0 1
et;
34
x11x63x6cx6dx50x5aΦ?1(s) =
1?s
0 1
e?s;x57x55x79Φ(t)Φ?1(s) =
1 t?s
0 1
et?s,
x73x74x4dx3x10x48x3bx7e,(2.2.10)x49x2 t0 = 0,x5cx9x11(2.2.10)x6c
’(t) = et
1 t
0 1
1
1
+
Z t
0
et
1 t?s
0 1
e?s
e?s
0
ds
=
t?1
1
et +et
Z t
0
e?2s
0
ds
=
t?1
1
et +
(et?e?t)=2
0
=
tet?(et +e?t)=2
et
:
x2ax73x17 x3x5cx10x48x45x46x66x79x77x6dx3ex2x76x7x59x9,x28x69x9x7x2x39x10xfx6dx17xd
x9x28,x52x76x33x2x6x2ax44x9x27xp = e?tv,x58x49v x9x6x2ax6ax36x2x8x77x26,x64x65x76
x33x6c,?e?tv = e?tAv + e?tv1,x73(?I?A)v = v1 x9x6c,v = (?1=2;0)T,x27
x63x6cx44x9xp = e?t(?1=2;0)T,x11x36x39 2.2.7,x27x9x27x = xp + Φ(t)c,x1bx11x7
x21x13x6bx3bx4ex36x14x74x8x77x26c = v2?xp(0) = (?1=2;?1)T.
35
x2.3 x56x2ex2fx30x31x1cx1dx19x1ax32
x57x66x6cx63x5cx3ax3bx76x33x9x2xex24x39x10x7bx4e,x59x9x3ax3bx76x33x2x63x7bx1ax5cx59x5ax58
x3xdx6ax5x76x33x2x2x9x25,x6d A(t) x1bx9x8x76x25x29,x59 x0 = A(t)x x2x2x9x25x3c
x4fx4cx4d; x3dx6d A x27x8x76x25x29,x7cx7dx7ax79x3ex3fx76x7x4fx17x59x5a x0 = Ax x2x6x64x69
x1dx33x30x2x9x25x4d.
x6x42 x2ax2bx56x23x60x7dx51x7ax11x73
x3x5c n xax45x8x76x25 A,xbx39x5c ex x2x57x1cx23x1ex20,x7cx7dx36x7
eAt · exp(At),= I +At+ 12!A2t2 +￠￠￠+ 1m!Amtm +￠￠￠ ; (2:3:1)
x58x49 I x27 n xax3ex2dx25,Am x27 A x2 m x5x57,x5fx59 A0 = I,0! = 1,(2.3.1) x67
x1x2x6bx25x1cx23x3x14x15x51x36x2t,x14x15nxax8x76x25Ax18x2.1x49x2x58x23x9x4ex4fx2.
x45x46x66,x59x7bjIj = 1,jAmj?jAjm,x8x1cx23
1+jAtj+ 12!jAj2jtj2 +￠￠￠+ 1m!jAjmjtjm +￠￠￠ = ejAjjtj
x27 (2.3.1) x67x1x2x23x1cx23,x57x55 (2.3.1) x67x1x4ex4f,x27x63 exp(At) x4ex45x9x6x2a n
xax45x76x25,x5ex73(2.3.1)x2x36x7x79x74x7.
x63x6,x3x14x6x2t 2 R,x1cx23(2.3.1)x3jtj? a,a > 0,x78x9x6x5dx4ex4fx2.
xfx6dx36x7(2.3.1),x6cx6dxdx7fx77x21x3bx43:
x14I) exp(?It) = exp(?t)I,x58x49?x27x8x23.
(II) x40nxax8x76x25Ax1fB x24x7x4fx7d,x73AB = BA,x8x79
exp((A+B)t) = exp(At)exp(Bt)); (2:3:2)
x44x9x31,x27x27?I x4ex14x15x57xax6bx25x24x7x4fx7d,x11(I),(II)x6c
exp((?I +N)t) = exp(?It)exp(Nt) = exp(?t)exp(Nt):
x57x58x3cx27x27Ax4e?Ax24x7x4fx7dx3cx11(I),(II)x6c
exp(At)exp(?At) = exp((A?B)t) = exp(0I) = I
x27x63
(exp(At))?1 = exp(?At); (2:3:3)
(III) x40x6bx25Ax2cx39x5cx6bx25J,A = TJT?1,x8x79
exp(At) = T exp(Jt)T?1,(2:3:4)
(VI) x40J x9x75x71x3x57x25,J = diag(J1;J2;:::;Js)x8
exp(Jt) = J = diag(exp(J1t);exp(J2t);:::;exp(Jst))
36
(V) x40 Jj x9 kj xa Jordan x7dx25 Jj =?jI + N,x27x27 Nkj = 0x3cx8x11x3bx43(II)
x1fx36x7(2.3.1),x6dx50
exp(Jjt) = exp(?jt)
I +tN + t
2
2!N
2 +￠￠￠+ tkj?1
(kj?1)!N
kj?1
:
x7dx4c 2.3.1 x40 A x27 n xax8x45x76x25,x8x6bx25x2ax23x22x23 Φ(t) = exp(At) x27
x76x33x0 = Axx2x33x30x45x2x9x25,x6x64x31x3cx1at = t0 x29x37x38x7x21x13x6bΨ(t0) = I
x2x2x9x25x27Ψ(t) = exp(A(t?t0)).
x6fx70 x45x46x66,x11 (2.3.1) x73x7b Φ(0) = exp(0) = I; x62 (2.3.1) x29x68x3 t
x59x73x6c
Φ0(t) = ddt[exp(At)]
= A+A2t+ 12!A3t2 +￠￠￠+ 1(m?1)!Amtm?1 +￠￠￠
= A[I +At+ 12!A2t2 +￠￠￠+ 1(m?1)!Am?1tm?1 +￠￠￠]
= Aexp(At) = AΦ(t):
x24x17 exp(At) x9x33x30x2x9x25x3cx3x5cx14x15 t,x30x9x39x1fx4fx25,x6x64x31 Ψ(t) =
exp(A(t?t0)) = exp(At)exp(?t0A) = Φ(t)exp(?t0A) x9x2x9x25x1fx39x1fx4fx25
x2x24x40x3cx18x36x392.2.5x65x9x2x9x25,x30x37x38x7x21x13x6b,x36x39xdx45.
x11x6bx25x2ax23x2x3bx43 (I)-(V) x7cx7dx4fx17x6cx4dx6bx25 exp(At) x2x6x69x59x7,x55x30
x4fx7bx6bx25exp(At)x2x67x54,x6d?j x9x45x23x29,x9x11x45x22x23tk exp(?jt),xc(x6d?j =
fi§ifl x9x60x23x29x3cx11Eulerx37x64exp((fi§ifl)t) = exp(fit)[cos(flt)§isin(flt)]
x4fx17x6cx4dx45x22x23) tk exp(fit)cos(flt),x1f tk exp(fit)sin(flt),j = 1;:::;s,k =
0;1;:::;kj?1x2x45x2fx23x2x3ax3bxex0x3c
x1ax42 exp(At) x60x58x59
x11x36x39 2.2.7 x1fx36x39 2.3.1 x73x7bx76x33 x0 = Ax x2x27x9x27 x = exp(At)c,
x58x49c x27x14x74 n x66x8x77x26; x55x7x2bx47x48x0 = Ax;x(t0) = x0 x2x9x27 x(t) =
exp(A(t?t0))x0,x60x67x28x28x59x9x2x63x7bx9x59x5a exp(At),x66x67x17x66x7cx7dx45x46x66
x7ax7x6cx4dx4dx59exp(At)x2x76x7x14x60x27x76x7x6x3ex3cx3dx9x28x76x7x4ex6fx66x50x6x2ax6bx25
T x3bx59x5aJordanx33x30x1 J,x55x6bx25 T x9x6bx25 A x2xcx7x67x44x31x77x26x61x2x27 T
x2x21x77x26x17x6x36x2x5x50xex72x2x3cx66x50x4fx3bx1ax77x61x1b,x1ax45x46x66exp(At)x2x66x50
x4fx7bx17x77x77x6dx3ex2x76x7x3bx6cx4d.
x13x73x17,x7cx7dx11x64x23x7bx29x4fx7b,x3x5cx14x6?x2nx5x3fxdx64pn(?),x6bx17m
x5x3fxdx64pm(?)x4fx17x6cx4dx6x2ax1bx5dx45q(?)x17xcx6x2ax5x23x70x5cmx2x5ex64r(?),
x4fx71x72x36x64:
pn(?) = pm(?)q(?)+r(?)
xbx39x31,xfx6dx29x75x71x39x10x3cx3x5cx3dx2ax51x36x2t,x4fx17x62x63x5c?x2x15x22x23e?t x71x72
x71x77x36x64:
e?t = pm(?)q(?;t)+c0(t)+c1(t)?+c2(t)?2 +￠￠￠+cm?1(t)?m?1; (2:3:5)
37
x58x49 q(?;t) x9? x2x15x22x23,cj(t),j = 0;1;￠￠￠ ;m? 1 x9 m x2ax7fx7bx2 t
x2x45x22x23x3cx6dx7ax7b pm(?) x24x79 s x2ax1bx57x2xf?j,(j = 1;2;￠￠￠ ;s) xcx58x3f
x23 kj x29,x4fx17x7bx6ax36x2fx23x7x2cx6x31x4ex36,x76x7x71x77,x62?j x64x65 (2.3.5) x64x6c
c0+c1?j+c2?2j +￠￠￠+cm?1?m?1j = e?jt,x6dkj > 1x29,x36x64x31x3x76x33(2.3.5)x29
x68x63x5c?x13x5x59x73kj?1x5xd,x26? =?j,x5fx4fx6ckj x2ax76x33:
8>
>><
>>>
:
c0 +c1?j +c1?2j +￠￠￠+cm?1?m?1j = e?jt;
c1 +2c2?j +￠￠￠+(m?1)cm?1?m?2j = te?jt;
￠￠￠
(kj?1)!ckj?1 +￠￠￠+(m?1)￠￠￠(m?kj +1)cm?1?m?kjj = tkj?1e?jt:
(2:3:6)
x3x5cpm(?)x2x3dx2axfx17x71x63x5x39,x5fx6cmx2ax76x33,x4fx17x4ex36mx2ax45x22x23cj(t),
j = 0;1;￠￠￠ ;m?1.
x71x13x24x79x2?j x9x45xf,x8(2.3.6)x9mx2ax45x76x33;x71x13x58x49x79?j x9x60xf,
x8x4fx62(2.3.6)x49x75x5ax45x4dx1fx60x4d,x24x6c 2kj x2ax45x76x33,x28x29,x3x5c?j x2x5fx61x60
xfx24x1bx7bx1bx68x4x76x33x4d (x27x27x3x5c?j x2x5fx61x60xfx2 kj x2ax76x33x16x5fx61x24x9x3
x5c?j x2kj x2ax76x33).
x11x3ax3bx64x23x39x10x7bx4e,x40pm(?)x27nxax6bx25Ax2x21x2x3fxdx64,x11x21x2x3fxd
x64x2x36x7
pm(A) = 0:
x36x64x66x6a(2.3.5)x49x2?x7bAx3bx64x7c,x6c
exp(At) = c0(t)I +c1(t)A+￠￠￠+ck?1(t)Am?1,(2:3:6)
x57x55x7x2bx47x48x0 = Ax,x(0) = x0 x2x9x24x9
x = exp(At)x0 = c0(t)x0 +c1(t)Ax0 +￠￠￠+cm?1(t)Am?1x0,(2:3:7)
x6x64x65x4fx17x16Ax2x44x31x3fxdx64det(?I?A)x27Ax2x21x2x3fxdx64,x3dx71x13A
x2xex70x3fxdx64 pm(?) x2x5x23 m x70x5c A x2xax23 n x29,x60x67x66x50x2x29x2ax3bx1fx66
x50x26x17x79x77x23x2x52x53.x3dx3x5cx17x77x29x69x44x45x75x76x3cx79x2fx6dx3ex2x44x45x76x7xc
1,x71x13 A x2x44x31x2bx17x8x5c?,x8x79x77x6dx3ex2x76x7x3bx6cx5a exp(At),x76x7x71
x77,x11x3bx43(2.3.2)
exp(At) = exp(?It+(AI)t) = exp(?It)exp((AI)t) = e?t exp((AI)t)
x11x5c(AI)n = 0,x24x17xfx6dexp(At)x2x36x7(2.3.1),x6c
exp(At) = e?t
n?1X
k=0
tk
k!(AI)
k,(2:3:8)
2,x71x13x56x6fx59x7x2bx47x48 dxdt = Ax,x(0) = vx2x9x3cx5fx30vx62x3dx9x6bx25A
x2x3xdx5cx44x31x2b?x2x44x31x77x26,(x73Av =?v ),x8x7x2bx47x48x2x9x27
x(t) = exp(At)v =
1X
k=0
tk
k!A
kv =
1X
k=0
(?t)k
k! v = e
tv,(2:3:9)
38
x7cx7dx3cx1ax17Ax9x1axax45x8x6bx25x2x75x76x27x7c,x52Ax2x29x2ax44x31x2bx9?1,?2.
x79x7ax3x5c?1,?2 x9Ax2x1bx2cx8x2x45x44x31x2bx2x75x36,x28x29xfx6d(2.3.6)x6c
‰ c
0(t) +?1c1(t) = e?1t
c0(t) +?2c1(t) = e?2t
x11x63x9x6c c0(t) = (?2e?1t1e?2t)=(?21);c1(t) = (e?2t?e?1t)=(?21),
x62x63x64x65(2.3.7)x6c
exp(At) = [(?2e?1t1e?2t)I +(e?2t?e?1t)A]=(?21),(2:3:10)
x69x0x5fx57xbxdx6cx6fx5cx59x45x2x3x60x2x36x64
exp(At) = e?1t(A2I)=(?12)+e?2t(A1I)=(?21),(2:3:10)?
x6d?1;2 = fi§iflx3c fl 6= 0x29x3cxfx6d(2.3.6)xcEulerx37x64xc
c0(t)+(fi+ifl)c1(t) = e(fi+ifl)t = efit (cos(flt)+isin(flt))
x26x66x64x49x45x4dx4ex60x4dx75x9x2cx8x3cx6c
c1(t) = efit sin(flt)=fl,c0(t) = efit [cos(flt)?fisin(flt)=fl],x73x6c
exp(At) = efit
(cos(flt)?fisin(flt)=fl)I + sin(flt)fl A
,(2:3:11)
x69x0x5fx57xbxdx6cx6fx5cx59x45x2x36x64
exp(At) = efit
cosfltI + sinfltfl (A?fiI)
,(2:3:11)?
x6d?1 =?2 = fi x29,x6bx4dx7bx66x38x6dx3ex37x64 (2.3.8) x59x6bx25x2ax23x6x3cx65x4fx17
x7b(2.3.6)x59x6c ‰
c0(t)+fic1(t) = efit;
c1(t) = tefit;
x11x63x9x6c c0(t) = (1?fit)efit;c1(t) = tefit,x64x65(2.3.5)x6c
exp(At) = efit [(1?fit)I +tA] = efit [I +t(A?fiI)]
x30x65x4fx17x1dx2(2.3.11)x49x26fl x17x42x2x29x2x8x6f.
x76 2.3.1 x52A =
2 1
3 4
,x59exp(At).
x2a det(?I?A) =
flfl
flfl2?1?34
flfl
flfl = (1)(5) = 0,x12x79?1 = 5,
2 = 1,x5cx9x11(2.3.10)*x6c
exp(At) = 14
‰
e5t
1 1
3 3
+et)
3?1
3 1
= 14
e5t +3et e5t?et
3(e5t?et) 3e5t +et
:
39
x76 2.3.2 x52 A =
1 1
2 3
,x59 exp(At).
x2a det(?I?A) =
flfl
flfl1?123
flfl
flfl =?2?4?+5 = 0,
x12x6c?1 = 2?i,?2 = 2+i,x73 fi = 2;fl = 1,x5cx9x11(2.3.11)*x6c
exp(At) = e2t
‰
cost
1 0
0 1
+sint
1 1
2 1
= e2t
cost?sint sint
2sint cost+sint
:
x76 2.3.3 x52 A =
2
4
4?1 0
3 1?1
1 0 1
3
5,x59 exp(At).
x2a Ax2x44x31x76x33x27
det(?I?A) =
flfl
flfl
flfl
4 1 0
31 1
1 01
flfl
flfl
flfl = (2)3 = 0;
x11x63x73x50Ax2x44x31x2bx4cx9? = 2,x27x63x11(2.3.8)x6c
exp(At) = e2t
8>
<
>:
2
4
1 0 0
0 1 0
0 0 1
3
5+t
2
4
2?1 0
3?1?1
1 0?1
3
5+ t22
2
4
2?1 0
3?1?1
1 0?1
3
5
29>=
>;
= e2t
2
4
1+2t+ t22?t?t2=2 t2=2
3t+t2 1?t?t2?t+t2
t+t2=2?t2=2 1?t+t2=2
3
5
x762.2.4 x59x9x7x2bx47x48 x0 = Ax =
2
4
2?1?1
3?2?3
1 1 2
3
5x;x(0) = x0 =
2
4
1
0
1
3
5
x2a x11x5cAx0 = x0,x24x17x11(2.3.9)x6c
x(t) = etx0
x76 2.3.5 x52 A =
2
4
4?1?1
1 2?1
1?1 2
3
5,x59x5ax76x33 x0 = Ax x2x33x30x2x9x25
exp(At).
40
x2a det(?I?A) =
flfl
flfl
flfl
4 1 1
12 1
1 12
flfl
flfl
flfl = (2)(3)2 = 0,x12x6c
1 = 3 x9x1ax3fxf,?2 = 2,x11x5cdet(A?3I) =
flfl
flfl
flfl
1?1?1
1?1?1
1?1?1
flfl
flfl
flflx2x63x8x5c1,x27
x63Ax2xex70x3fxdx64x27p2(?) = (2)(3)x57x55exp(At) = c0(t)I +c1(t)A,
x11(2.3.6)x64x6cc0(t) = 3e2t?2e3t,c1(t) = e3t?e2t,x2bx55x6c
exp(At) = e2t(3I?A)+e3t(?2I +A)
2
4
e2t +2e3t e2t?me3t e2t?e3t
e2t +e3t e2t e2t?e3t
e2t+e3t e2t?e3t e2t
3
5
x6bx42x23x79x7ax13x14x24x60x48x2a
x3x5cx39x6ax5x76x33x7x2bx47x48x2x59x9,x7ax11x8x23x20x6dx37x64 (2.2.10) x6x5a,x3dx6d
Ax27x8x76x25x29,x7x2bx47x48
x0 = Ax+f(t); x(t0) = x0
x2x9(2.2.10)x4fx17x59x7bexp(At)x2x3bx43(II)x2(2.3.3)x71x72
’(t) = eA(t?t0)x0 +
Z t
t0
eA(t?s)f(s)ds,(2:3:12)
x76 2.3.6 x59x5ax7x2bx47x48
x0 =
4?3
2?1
x+
sint
2cost;x(0) =
1
1
x2x9.
x2a x79x7ax59x5aAx2x44x31x2b
det(?I?A) =
flfl
flfl4 3?2?+1
flfl
flfl = (1)(2) = 0;
x11x63x73x6c?1 = 1,?2 = 2,x11x76x7x1ax6c exp(At) = c0(t)I + c1(t)A,x58x49
c0(t) = 2et?e2t,c1(t) = e2t?et,x64x65x6c
exp(At) =
3et?2et 3(et?e2t)
2(et?et) 3et?2e2t
x57x55x11(2.3.12),x17xc
Z t
0
ea? sin?d? = [eat(asint?cost)+1]=(1+a2)
41
x1f Z
t
0
ea? cos?d? = [eat(acost+sint)?a]=(1+a2);
x73x6c
’(t) =
3e2t?2et 3(et?e2t)
2(e2t?et) 3et?2e2t
1
1
+
Z t
0
3e2(t?s)?2et?s 3(et?s?e2(t?s))
2(e2(t?s)?et?s) 3et?s?2e2(t?s)
sins
2coss
ds
=
3e2t?3et +cost?2sint
2e2t?3et +2cost?2sint
:
x73,x1ax7c16x49x4fx50x7b(2.3.12)x59x44x9x6fx2bx4bx77x7ex2x40x75x20x50,x1ax41x7x4dx25xa
x3ax3bx8x2fx23x39x6ax5x76x33x2x59x9x76x7xd,x4fx17x7bx6x69x56x7bx59x73x1fx64x23x20x50x2x77x6d
x3ex2x76x7x59x28xbx39x6ax5xdx2x3ax3bx8x2fx23x39x6ax5x76x33x2x44x9 (x50x2.4,x65,xexd
x2x73).
42
x2.4 x33x34x30x31x1cx1dx19x1a
x7cx7dx1ax2.1x49x19x65x1fx7ax7x6anxax3ax3bx74x75x76x33(2-12)x21x72x33x30x3ax3bx76x33
x0 = A(t)x+f(t)x2x36x64,x27x63x24x79x63x5cx3ax3bx76x33x2x39x10x17x1x7bx5cx25xax3ax3bx76
x33,x3dx11x5cx76x33x2x44x45x3b,x6x76x6cx76x33x2x6x64x39x10x1fx76x7xdx7bx4dx76x33x66x29,x58
x72x3cx36x64x24x3cx79x1bx57x5cx6x64x36x64x2x7ex44x4;x4ax6x76x6cx58x67x3ax1ax6x72x1bx57x5cx6x64
x3ax3bx76x33,x55x9x25xax3ax3bx76x33x10x25x24x44x79x2x39x10x1fx9x48x76x7; x28x24x9x10x3x6fx34
x10x2x29x2ax6ex6fx76x6cx2x47x48.
x6x42 x4x1ax4cx1d
x7cx7dx6ax6x64nxax3ax3bx74x75x76x33x3fx71x71x77
x(n) +a1(t)x(n?1) +￠￠￠+an?1(t)x0 +an(t)x = f(t); (2:4:1)
x58x49 aj(t);j = 1;￠￠￠ ;n,xc f(t) x17x1ax5dx13 a? t? b x66x6ex6f,x5fx60x6df(t) · 0
x29x2nxax3ax3bx76x33
x(n) +a1(t)x(n?1) +￠￠￠+an?1(t)x0 +an(t)x = 0 (2:4:2)
x27(2–56)x24x3xdx2x22x11x79x7ax13x14,x60f(t)x27x23x79x7axb.
x77x6cx7cx7dx6ax2.1 x49x63x5cx3ax3bx76x33x6x64x39x10x2x3x4x1fx36x39x1bx14x7ex7fx1fx1bx14
xdx7fx31x18x19x25xax76x33x10x25x2x36x64x2bx4bx38x21x1fx7fx38,x65x66x1cx1dx6ax30x7dx1bx21x72x76x33
xex3xdx2x3x4x1fx36x39x14x17x39x9x1fxdx7f.
x7dx4c 2.4.1 x40 aj(t);j = 1;￠￠￠ ;n; x1f f(t) x1a [a;b] x66x6ex6f,x8x3 8t0 2
[a;b]x1fnx2ax23x00;x01;￠￠￠ ;x0n?1,x76x33(2.4.1)x3ax1ax2cx6x37x38x7x21x13x6b
x(t0) = x00;x0(t0) = x01;￠￠￠ ;x(n?1)(t0) = x0n?1
x2x9x = ’(t),x30x1ax15x2ax5dx13[a;b]x66x17x79x36x7.
x3x5cx3ax3bx6ax5x76x33(2.4.2),x3ex14x2bx39x72x4xc
x7dx4c2.4.2 x71x13 x1(t),x2(t),￠￠￠,xk(t) x9 (2.4.2) x2 k x2ax9,x8x3x14x6
x2kx2ax8x23c1,￠￠￠,ck,x3ax3bxex0c1x1(t)+c2x2(t)+￠￠￠+ckxk(t)x65x9(2.4.3)
x2x9.
x3x5cx36x7x1ax5dx13 [a;b] x66x2k x2ax23x2bx22x23x1(t),x2(t),￠￠￠,xk(t),x71x13x3a
x1akx2ax1bx4cx27x2x2x8x23c1,c2,￠￠￠,ck,x51x6cx3t 2 [a;b]x79c1x1(t)+c2x2(t)+
￠￠￠+ckxk(t) · 0,x8x60x28k x2ax22x23x1a[a;b]x66x22x11x47x77,x79x8x24x60x30x7dx22x11x40
x77.
x52x1(t),x2(t),￠￠￠,xn(t)x9nx2ax1a[a;b]x66(k?1)x5x6ex6fx4fx74x2x23x2bx22
x23,x31x63x11x30x7dx2x72x2x4bx21x64
W[x1(t);x2(t);￠￠￠ ;xn(t)] · W(t),=
flfl
flfl
flfl
flfl
fl
x1(t) x2(t) ￠￠￠ xn(t)
x01(t) x02(t) ￠￠￠ x0n(t)
...,..,..
x(n?1)1 (t) x(n?1)2 (t) ￠￠￠ x(n?1)n (t)
flfl
flfl
flfl
flfl
fl
43
x60x27x28x72x22x23x2Wronsky (Wronsky)x44x5cx13.
x7dx4c2.4.3 x71x13x22x23x1(t),x2(t),￠￠￠,xn(t)x1a[a;b]x66x3ax3bx2cx63,x8x1a
[a;b]x66x30x7dx2Wronskyx4bx21x64W(t) · 0.
x7dx4c2.4.4 x71x13(2.4.2)x2x9x22x23x1(t),x2(t),￠￠￠,xn(t)x1a[a;b]x66x3a
x3bx4bx63,x8x30x7dx2Wronskyx4bx21x64x1at 2 [a;b]x66W(t) 6= 0.
x7dx4c2.4.5 nxax3ax3bx6ax5x76x33(2.4.2)x58x36x3ax1anx2ax3ax3bx4bx63x9.
x7dx4c 2.4.6 x71x13 x1(t);x2(t);￠￠￠ ;xn(t) x9 (2.4.2) x2 n x2ax3ax3bx4bx63x9,
x8x30x2x27x9x27x = c1x1(t)+c2x2(t)+￠￠￠+cnxn(t),x58x49c1;c2;￠￠￠ ;cn x27x14
x74x8x23;x55x30x28x2ax27x9x3ax7ex4d(2.4.2)x2x6x45x9.
x1cx1d2.2.1 (2.4.2)x2x3ax3bx4bx63x9x2xex23x2ax23x27n.
x5cx9x79x7ax22x11x13x14(2.4.2)x60x2ax67x48x51x4x5nx34x22x11x52x4a.x7cx7dx60(2.4.2)
x2x14x15nx2ax3ax3bx4bx63x9x27x30x2x6x2ax7fx52x2ax24.
x17x9x23x79x7ax22x11x13x14
x7cx7dx3cx1ax68x4dx3ax3bx39x6ax5x76x33 (2.4.1),xbx39x5cx6ax5x1fx39x6ax5x76x33x9x52x13
x2x63x2f,x7cx7dx65x79
x51x252.4.1 x40 ’(t) x27 (2.4.1) x2x9,?(t) x27 (2.4.2) x2x9,x8 ’(t) +?(t)
x27(2.4.1)x2x9.
x51x252.4.2 x40’(t)x1f?(t)x17x9(2.4.1)x2x9,x8x20’(t)(t)x27(2.4.2)
x2x9.
x7dx4c2.4.7 x52 ’1(t);’2(t);￠￠￠ ;’n(t) x27x6ax5x76x33 (2.4.2) x2x2x10x9xe,
’(t)x27x39x6ax5x76x33(2.4.1)x2x14x6x44x9,x8x39x6ax5x76x33(2.4.1)x2x27x9x27
x = c1’1(t)+c2’2(t)+￠￠￠+cn’n(t)+’(t); (2:4:3)
x58x49c1;c2;￠￠￠ ;cn x27x14x74x8x23;x30x27x9(2.4.3)x3ax7ex4d(2.4.1)x2x6x45x9.
x11x63x73x50,x27x4dx6cx4dx27x9(2.4.3),x6x9x4cx5ax58x3xdx6ax5x76x33(2.4.2)x2x2x10
x9xe,x1ax9x59x5ax39x6ax5x76x33 (2.4.1) x2x6x2ax44x9; x3dx9x59x7bx8x23x20x6dx7,xdx1dx4f
x17x7x11x29x1dx2bx4bx6x5x40x75x6cx4d.
x45x46x66,x40x6a (2.4.1) x71x72 (2.1.11) x36x64x2x3ax3bx76x33,x8x11 (2.4.2) x2x2x10
x9xe’1(t);￠￠￠ ;’n(t) x73x7bx58x3xdx6ax5x76x33x2x2x10x9x6bx25x27
Φ(t) =
2
66
64
’1(t) ￠￠￠ ’n(t)
’01(t) ￠￠￠ ’0n(t)
...,..,..
’n?11 (t) ￠￠￠ ’n?1n (t)
3
77
75,(2:4:4)
x5cx9x11 (2.2.10) x64x73x4fx1dx5ax22x23x77x26 Rtt0 Φ(t)Φ?1(s)ef(t)ds x2x19x6x2ax75x26x24
x9 (2-69) x2x6x2ax44x9 ’p(t),x28x28 f(t) = (0;:::;0;f(t))T,x17 k(t;s) x59
Φ(t)Φ?1(s) x2x19x6x4bx19 n x21x2x67x54,x73 (2.4.4) x19x6x4bx77x26x4e Φ?1(s) x19 n
x21x77x26x2x23x26x40; x27x63x2x27 t x2x22x23,x30x9 (2.4.2) x2x9; x55 Φ?1(s) x19 n x21x77
44
x26x27 Φ(s) x19 n x4bx2x64x23x5ex7dx64x4e 1W(s) x2x24x40,x28x28 W(s) x27 (2.4.4) x2x4b
x21x64;x27x63k(s;t)x4fx17x71x72
k(s;t) = 1W(s)
flfl
flfl
flfl
flfl
flfl
fl
’1(s) ’2(s) ￠￠￠ ’n(s)
’01(s) ’02(s) ￠￠￠ ’0n(s)
...,..,..,..
’(n?2)1 (s) ’(n?2)2 (s) ￠￠￠ ’(n?2)n (s)
’1(t) ’2(t) ￠￠￠ ’n(t)
flfl
flfl
flfl
flfl
flfl
fl
,(2:4:5)
x5cx9(2.4.1)x2x27x9x27
x = c1’1(t)+￠￠￠+cn’n(t)+
Z t
t0
k(s;t)f(s)ds; t0 2 [a;b],(2:4:6)
x76 2.4.1 x59x5ax76x33tx00?x0 = t2 x2x27x9,(x73x74,x28x28f(t) = t)
x2ax73x4 x3xdx2x6ax5x76x33x27tx00?x0 = 0,x5cx9x79
d
dt lnx
0 = x00
x0 =
1
t =
d
dt lnt
x11x63x9x6c x0 = 2c2t,x57x55x27x9x27 x = c1 + c2t2,c1,c2 x27x14x74x8x23,x16x63x6ax5
x76x33x2x2x10x9xex271,t2,x27x63x11x142.4.5x3ex6ck(s;t) =?s2 + 12st2,x5cx9
’(t) =
Z t
0
s2 + t
2
2s
sds =?t
3
6 +
t3
2 =
t3
3,
x12x2bx76x33x2x27x9x27x = c1 +c2t2 + t
3
3,
x2ax73x17 x52x39x6ax5x76x33x2x44x9x2x36x64x27xp = ct3,x64x65x76x33x6cc = 13,x52x3
xdx2x6ax5x76x33x2x9x3cx79x36x64x = tn,x64x65x3xdx2x6ax5x76x33x59x6cn = 0x69n = 2,
x27x63x11x36x392.4.7,x6cx2bx39x6ax5x76x33x2x27x9x27x = c1 +c2t2 + t
3
3,
x64x9x32x22x23x22x11x13x14
x27x4dx59x5anxax3ax3bx76x33x2x27x9,x56x6fx59x5ax58x3xdx6ax5x76x33x2nx2ax3ax3bx4bx63
x9;x4fx9x3x5cx6x64x20x2fx23x6ax5x76x33x3bx7e,x69x16x0x4bx6x64x2x76x7x3bx59x58x2x10x9xe.
x3dx3x5cnxax8x2fx23x3ax3bx6ax5x76x33,x4ax56x6fx27x32x9x6x2ax64x23x76x33,x6ex40x75x20x50x17
x2ax79x58x6f,x73x4fx59x5ax58x2x10x9xe,x55x6dx39x6ax5xdx27x28x6xbx22x23x29,x65x56x6fx59x7b
x64x23x20x50x4ex59x73x20x50x24x4fx17x59x5a n xax8x2fx23x3ax3bx39x6ax5x76x33x2x27x9; x28x24x9
x8x2fx23x3ax3bx76x33x2x6ex6fx44x4.
1,x1x5cx5dx60x2x4ax49x23 x40x45x20x26tx1ax5dx13[a;b]x66x20x21x29,x58x22x23x2bz(t)
x9x16x29x23x2b,x73 z(t) = ’(t) + i?(t),x28x28 ’(t);?(t) x5dx27x45x2bx22x23,x8x60 z(t)
x27x1x5cx5dx60x2x4ax49x23,x63x5c z(t) x2x8x6fx42x6ex6fx42x59x73x1fx40x75x17x9x75x9x3 ’(t)
45
x1f?(t)x2bx4bx2,x27x63x18x4cx4ex45x20x26x2x45x2bx22x23x2cx57.x7cx71x40z(t);z1(t);z2(t)x5d
x27x29x2bx22x23,cx27x29x8x23,z(t) = ’(t)+i?(t),x8x79
z0(t) = ’0(t)+i?(t);
[z1(t)+z2(t)]0 = z01(t)+z02(t);
[cz(t)]0 = cz0(t);
[z1(t)z2(t)]0 = z01(t)z2(t)+z1(t)z02(t);
x30x6d(z2(t) 6= 0)x29,x79
[z1(t)=z2(t)]0 = [z01(t)z2(t)?z1(t)z02(t)]=z22(t):
x4ax6,x29x6cx47x4dx2Eulerx37x64
e(fi+ifl)t = efit(cosflt+isinflt);
x1ax77x6cx62x7x8x7bx4d.x63x6x40 k1;k2 x5dx27x29x23,k x27x29x8x23,x8x79
ek1tek2t = e(k1+k2)t;
(ekt)0 = kekt;
dn
dtne
kt = knekt;
x28x72x17x4ex45x2bx2x2ax23x22x23x2x20x50x6x58.
x59x7bx7x4x5x2x23x12x13x8,x1x1ax7ax3x1ax45x12x21x5ex45x47x12x60x62x7e,x4fx17x53x54x4d
xdx77x21x29x13x36x39.
x7dx4c2.4.8 x71x13 (2.4.2) x49x24x79x2fx23 aj(t);(j = 1;￠￠￠ ;n); x17x9x45x2bx22x23,
x55x = z(t) = ’(t)+i?(t) x9x30x2x29x2bx9,x8z(t)x2x45x4d ’(t)x42x60x4d?(t)x17
xcz(t)x2x5fx61x29x2bx22x23z(t)x65x17x9x30x2x9.
x7dx4c2.4.9 x40x3ax3bx39x6ax5x76x33
x(n) +a1(t)x(n?1) +￠￠￠+a(n?1)(t)x0 +an(t)x = u(t)+iv(t)
x79x29x2bx9 x = U(t)+iV(t),x58x49 aj(t);j = 1;￠￠￠ ;n;u(t);v(t);U(t);V(t) x5d
x27x45x2bx22x23,x8x9x2x45x4d U(t) x1fx60x4d V(t) x75x9x27x76x33
x(n) +a1(t)x(n?1) +￠￠￠+an?1(t)x0 +an(t)x = u(t)
x1fx76x33
x(n) +a1(t)x(n?1) +￠￠￠+an?1(t)x0 +an(t)x = v(t)
x2x9.
2,nx19x32x22x23x22x11x79x7ax13x14 x51x52x1ax6ax5x76x33(2.4.2)
x(n) +a1x(n?1) +￠￠￠+an?1x0 +anx = 0; (2:4:7)
46
x49 a1;￠￠￠ ;an x5dx27x45x8x23,x3x5cx28x2ax8x2fx23x3ax3bx6ax5x76x33,x7cx7dx57x58x4fx17x6ax30
x21x72x8x2fx23x6ax5x76x33xex2bx4bx59x9,x1bx32x3x5cx76x33xex2x59x9,x6d n > 3x29,x59x9
x32x33x24x77x27x29x2a,x3dx3cx1ax7cx7dx62x3x6x64nxax76x33x2bx4bx34x10,x58xex13x4ex2x27x76x33
xex2bx4bx34x10x29x18x4cx6x58,x3dx76x7x62x2fx27x53x54x1fx6dx3e,x28x18x4cx9x11x5cx76x33x2x44x45
x36x64x24x4dx36x2.
x59x9(2.4.7)x2x63x7bx9x4cx5ax30x2x2x10x9xe,x73x30x2nx2ax3ax3bx4bx63x9,x4ex34x10
x76x33xex29x6x58,x3x5c(2.4.7)x7cx7dx65x2dx59x58x36x71x = e?tp(t)x2x9,x58x49p(t)x27
x6ax36x2tx2x3fxdx64,?x27x6ax36x46x23.
x36x7x74x75x50x7dDx71x77:
Dx = dxdt; Dm+1x = D(Dmx) = d
m+1x
dtm+1 ; m 2 N
x19x36D0x = x.
x5cx9x8x2fx23nxax6ax5x76x33(2.4.7)x4fx17x71x72x50x7dx76x33x36x64,L(D)x = 0,x58
x49
L(?),=?n +a1?n?1 +￠￠￠+an?1?+an,(2:4:8)
x60x27 (2.4.7) x2x2ex67x68xbx13 x44x31x3fxdx64 (2.4.8) x2xfx60x27 (2.4.7) x2x2ex67x69.
(2.4.8)x4ex62 (2.4.7) x21x72x76x33xe x0 = Ax x52xdx2x44x31x3fxdx64 det(?I?A) x9
x6x58x2.
x52(2.4.8)x2x2bx1bx2cx8x2xfx27?j,j = 1;￠￠￠ ;s,?j x2x3fx23x27kj,x8x50x7dx76
x33L(D)x = 0x4fx17x71x72
sY
j=1
(Dj)kjx = 0:
x27x63,x40xx9x50x7dx76x33
(Dj)kjx = 0 (2:4:9)
x2x9,x8x4fx50xx65x9(2.4.7)x2x9.
x1ax59x9x50x7dx76x33(2.4.9)x52x29,x7ax6x5ax6x2ax74x75x37x64,x52g(t)x9tx2x4bx6fx5
x4fx74x22x23(x71x3fxdx64),x8x4fx17x7bx23x41x75x22x7xdx7fx72x4
Dm
e?tg(t)
·
= e?t (D+?)m g(t); m 2 N
x57x55x3x5cDx2x3fxdx64L(D)x72x4
L(D)e?tg(t) = e?tL(D+?)g(t):
x27x63,x52(2.4.9)x2x9x2x36x64x27x = e?jtg(t),x11x66x64x6c:
(Dj)kjx = e?jtDkjg(t) = 0:
x27x63,g(t)x4fx17x16x271,t,t2,￠￠￠,tkj?1,x24x17(2.4.7)x79kj x2ax9:
tke?jt; k = 0;￠￠￠ ;kj?1;
47
x6d?j x9x45x23x29,x30x7dx9(2.4.7)x2kj x2ax45x9;x6d?j = fij +iflj x9x60x23x29,x30x7d
x9x45x2fx23x76x33(2.4.7)x2x29x2bx9,x27x63xfx6dx36x39III,x30x7dx2x45x4dx4ex60x4d
tkefijt cos(fljt); tkefijt sin(fljt); k = 0;1;￠￠￠ ;kj?1:
x9(2.4.7)x22kj x2ax9,x3x3dx6x2ax45x2x44x31x2bx4ex60x2x6x3x5fx61x29xfx49x2x14x6x2a
x17x71x63x5x39,x24x6cx4d n x2ax9,x4fx17xdx7fx30x7dx9x3ax3bx4bx63x2,x57x55x30x7dx2x3ax3bxe
x0x24x9x76x33(2.4.7)x2x27x9.
x76 2.4.2 x59x5ax76x33 x(4)?x = 0 x2x27x9
x2a x44x31x76x33x27?4? 1 = 0,x27x63x30x2x6bx2axfx27?1 =?1,?2 = 1,
3 = i,?4 =?i,x30x7dx3xdx6bx2ax3ax3bx4bx63x2x45x9 e?t;et;cost;sint,x27x63x2bx76
x33x2x27x9x27
x = c1e?t +c2et +c3 cost+c4 sint;
x58x49 c1;c2;c3;c4 x27x14x74x45x8x23.
x762.4.3 x59x9x76x33x000 +x = 0
x2a x44x31x76x33x27?3 +1 = 0,x58xfx27?1 =?1;?2;3 = 1=2§ip3=2,x4ex63
x3xdx2x65x2ax3ax3bx4bx63x9x27 e?t; e12t cos
p3
2 t; e
1
2t sin
p3
2 t,x27x63x2bx76x33x2x27x9
x27 x = c1e?t +(c2 cos
p3
2 t+c3 sin
p3
2 t)e
1
2t,x58x49 c1;c2;c3 x27x14x74x45x8x23.
x76 2.4.4 x59x9x76x33 x000?3x00 +3x0?x = 0.
x2a x44x31x76x33x27?3?3?2 +31 = (1)3 = 0,x73? = 1 x27x65x3fxf,
x12x58x27x9xdx27 x = (c1 +c2t+c3t2)et,x58x49 c1;c2;c3 x27x14x74x45x8x23.
x76 2.4.5 x59x5ax76x33 x(4) +2x00 +x = 0 x2x27x9.
x2a x44x31x76x33x27?4 +2?2 +1 = (?2 +1)2 = 0,x73? = §i x27x1ax3fxf,x27
x63x3xdx2x6bx2ax45x9x27 cost; tcost; sint; tsint,x57x55x27x9x27
x = cost(c1 +c2t)+sint(c3 +c4t);
x58x49 cj;j = 1;￠￠￠ ;4; x27x14x74x45x8x23.
3,x4x4a(Euler)x13x14,x24x25Eulerx13x14x24x9x71x77x2x20x2fx23x76x33
tnx(n) +a1tn?1x(n?1) +￠￠￠+an?1tx0 +anx = 0; (2:4:10)
x58x49 a1;￠￠￠ ;an x5dx27x45x8x23,x1a(2.4.10)x49x2x25x20x26 t x2x20x26x7cx7d
jtj = e?; x73? = lnjtj; d?dt = t?1 (2?82)
x36x7x74x75x50x7dD = dd? x8x4fx17x7bx23x41x75x22x7xdx7fx17x77x63x2fx64:
dx
dt = t
1Dx;￠￠￠ ; dmx
dtm = t
mD(D?1)(D?2)￠￠￠(D?(m?1))x:
48
x5cx9x6a(2.4.10)x21x72x6x2ax17? x27x25x20x26x42x17 x x27x7fx7bx22x23x2 n xax8x45
x2fx23x3ax3bx6ax5x76x33,x5cx9x7ex76x33x2x = e x36x64x2x9x24x3xdx4d(2.4.10)x36x64x27
x = (§t)? x2x9,x7ex76x33x2x44x31x76x33x27
(1)￠￠￠(n+1)+a1?(1)￠￠￠(n+2)+￠￠￠+an?1?+an = 0,(2:4:11)
x65x60x27x9Eulerx76x33(2.4.10)x2x44x31x76x33,x71x13x76x33(2.4.11)x79 n x2ax1bx57x45
xf?1;￠￠￠ ;?n,x31x63x4fx17x6cx4d(2.4.10)x2x2x10x9xe(§t)?1,(§t)?2,::,(§t)?n.
x40x58x49x2 e? x27 k x3fxf,x8x6fx7b (§t)e?; (§t)e? lnjtj;￠￠￠ ; (§t)e? lnk?1jtj x64x7c
k x2a (§t)e?,x71x13 (2.4.11) x5ax3cx5fx61x29xf fi § ifl,x8x6fx7b (§t)fi cos(fllnjtj)
x1f (§t)fi sin(fllnjtj) x64x7cx29x2bx9 (§t)fi+ifl x1f (§t)fi?ifl,x40 fi§ifl x27 k x3f
xf,x8x4fx7bx71x77 2k x2ax45x2bx9x64x7c 2k x2ax2 (§t)fi§iflxc
(§t)fi cos(fllnjtj);(§t)fi ln(§t)cos(fllnjtj);￠￠￠ ;(§t)fi lnk?1jtjcos(fllnjtj);
(§t)fi sin(fllnjtj);(§t)fi lnjtjsin(fllnjtj);￠￠￠ ;(§t)fi lnk?1jtjsin(fllnjtj);
x3 (2.4.11) x24x79x45x3fxfx1fx29x3fxfx17x2bx4bx28x58x5x39xd,xexdx6cx4d (2.4.10) x2 n
x2ax3ax3bx4bx63x45x2bx9,x73x74,x6d t < 0 x29,x40 tfi x79x74x7,x8x1ax27x9x49 (§t)fi x4fx71
x72tfi x2x36x64.
x76 2.4.6 x59x5ax76x33 t2x00?tx0 +x = 0 x2x9.
x2a x28x9Eulerx76x33,x58x44x31x76x33x27
(1)+1 =?2?2?+1 = (1)2 = 0;
x12? = 1 x27x1ax3fxf,x58x3xdx5cx2bx76x33x2x29x2ax3ax3bx4bx63x9x27 t x1f tlnjtj,x27x63
x2bx76x33x2x27x9x27 x = t(c1 +c2 lnjtj); c1;c2 x27x14x74x45x8x23.
x76 2.4.7 x59x9x76x33 t2x00 +3tx0 +5x = 0.
x2a x28x9Eulerx76x33,x58x44x31x76x33x27?(1)+3?+5 = 0,x73
2 +2?+5 = 0,x58xfx27? =?1§2i;x27x63x2bx76x33x2x27x9x27
x = t?1(c1 cos(2lnjtj)+c2 sin(2lnjtj))
x58x49 c1,c2 x27x14x74x45x8x23.
4,x23x79x7ax13x14x60x2a x1ax10x3x2x19x1ax1f,x7cx7dx4fx17xfx6d(2.4.2)x2x2x10x9xe
x59x7bx8x23x20x6dx37x64x7x32x40x75x59x5ax39x6ax5x76x33(2.4.1)x2x6x2ax44x9.
x762.4.8 x59x5ax76x33x00 +x = 1cost x2x27x9.
x2a x79x7ax59x5ax3xdx6ax5x76x33x00+x = 0x2x2x10x9xe,x58x44x31x76x33x27?2+1 =
0,x11x63x73x6c x00 + x = 0 x2x2x10x9xex27 cost,sint,x11(2.4.6) x6c k(s;t) =
sintcoss?costsins,x24x17
’(t) =
Z t
0
k(s;t)f(s)ds
=
Z t
0
[sintcoss?costsins] 1cossds
= tsint+costlnjcostj;
49
x27x63x2bx76x33x2x27x9x27
x = costln(c1 cost)+(c2 +t)sint:
x3dx3x5cx8x45x2fx23x3ax3bx39x6ax5x76x33x3bx7e,x71x13x39x6ax5xd f(t) x79x71x77x29x68xb
x13x2x36x64
f(t) = e?0tpm(t);x69x6d?x27x60x23x29f(t) = Re
e?0tpm(t))
·
(2:4:12)
x58x49x6d?0 x9x45x8x23x29x3c pm(t)x27tx2mx5x45x2fx23x3fxdx64,x6d?0 x9x60x8x23x29x3c
pm(t)x27tx2mx5x29x2fx23x3fxdx64,x8x4fx17x7bx1ax77x2fx23x7x59x5a(2.4.1)x2x6x2ax44
x9,x67xdx14x66x6ax5x76x33x2x27x9x24x6cx4dx39x6ax5x76x33x2x27x9.
x3cx3fx31x7e,x6d?0 x9x45x23x3cx3x5c(2.4.12)x36x64x2x39x6ax5xd f(t),x7cx7dx59x39x6a
x5x76x33(2.4.1)x71x77x36x64x2x44x9
xp = e?0tq(t);
x58x49q(t)x27tx2x3fxdx64,x64x65(2.4.1)x2x50x7dx76x33(2.4.8)x6c
L(D)xp = e?0tL(D+?0)q(t) = e?0tpm(t);
x73
L(D+?0)q(t) = pm(t),(2:4:13)
x6d?0 x1bx9x44x31x76x33x2xfx29,L(?0) 6= 0,x24x17q(t)xdx27mx5x45x2fx23x3fxdx64,x6d
0 x9x44x31x76x33x2 k x3fx45xfx29,L(? +?0) x79x27x7d?k,x24x17 L(D +?0) x79x27x7d
Dkx3cx73(2.4.13)x0x68x49x2xex53xax73x23x2xax23x9k,x24x17x11(2.4.13)x4fx7bDkq(t)
xdx27mx5x45x2fx23x3fxdx64x3cx1fx5x40x75k x5x3cx3dx5x2x40x75x8x23x17x16x27x2x3cx8q(t)
x4fx17x16x72x36x64q(t) = tkrm(t)x3cx58x49rm(t)x9tx2mx5x45x2fx23x3fxdx64.
x6d?0 x9x60x23x29x3cx1f?0 x29xbx39x76x7x7ax59x5ax39x6ax5x76x33
L(D)z = e?0tpm(t)
x2x29x2bx44x9zpx3cx67xdxdx7bx36x392.4.9,xp = Rezp x24x9x2bx39x6ax5x76x33x2x44x9,x27
x63xp x2x36x64xdx27Re?e?0ttkrm(t)￠,x58x49rm(t)x9tx2x29x2fx23x2mx5x3fxdx64.
x76 2.4.9 x59x5ax76x33 x00?2x0?3x = 3t+1 x2x27x9.
x2a x79x7ax59x3xdx6ax5x76x33 x00? 2x0? 3x = 0 x2x27x9,x28x29x44x31x76x33x27
2?23 = 0,x11x63x4fx6cx44x31xfx27?1 =?1;?2 = 3,x12x3xdx6ax5x76x33x2x27x9
x27 x = c1e?t + c2e3t,x39x6ax5xdx9 f(t) = 3t + 1 x73?0 = 0 x1bx9x44x31xf,x27x63
x2bx76x33x2x44x9xdx16x72x6x5x3fxdx64xp(t) = At+B,x62x63x64x65x2bx76x33x6c
2A?3At?3B = 3t+1;
x1ax77x29x68x63x5c t x2x57x5x57x2x2fx23x79?3A = 3;?2A? 3B = 1,x11x63x59x6c
A =?1,B = 1=3;x27x63x2bx76x33x2x27x9x27x = c1e?t +c2e3t?t+1=3.
x762.4.10 x59x5ax76x33 x(3) +3x00 +3x0 +x = e?t(t?5) x2x27x9.
50
x2a x44x31x76x33x27?3 + 3?2 + 3? + 1 = (? + 1)3 = 0,x73? =?1 x27x65x3f
xf; x11x5cf(t) = (t?5)e?t,x73x50?0 =?1 x9x44x31x76x33x2x65x3fxf,x12x16x44x9x27
xp = t3(At+B)e?t,x62x63x64x65x2bx76x33x6c
L(D)xp = (D+1)3xp = e?tD3t3(At+B) = e?t(6B +24At) = e?t(t?5)
x11x63x9x6c A = 1=24,B =?5=6,x12x2bx76x33x2x27x9x27
x = e?t?c1 +c2t+c3t2 +t3(t?20)=24￠;
x58x49,c1;c2;c3 x27x14x74x45x8x23.
x76 2.4.11 x59x5ax76x33 x00 +4x0 +4x = cos(2t) x2x27x9.
x2a x44x31x76x33x27?2+4?+4 = (?+2)2 = 0,x5cx9? =?2 x27x1ax3fxf,x39x6a
x5xd f(t) = cos(2t) = Re(e2it),x272ix1bx9x44x31x76x33x2xfx21x27x63xdx16x2bx76x33x2x44
x9x2x36x64x27xp = Re(e2itc),x58x49cx9x29x8x23,x24x17xp = Acos(2t)+Bsin(2t),
x62x63x64x65x2bx76x33x6c
8Bcos(2t)?8Asin(2t) = cos(2t);
x75x9x1ax77 cos(2t) x1f sin(2t) x29x68x3xdx2x2fx23x73x6c A = 0,B = 1=8,x27x63x2b
x76x33x2x27x9x27
x = (c1 +c2t)e?2t +sin(2t)=8;
x58x49c1;c2 x27x14x74x45x8x23.
x3x5cx7c 2.4.11 x2x44x9 xp x78x79x71x77x9x7,x3x5cx5x6x29x23x20x50x2x1cx1dx9x77x7
x8x2x76x7.
x2a x11x5ccos(2t) = Re(e2it),x12x52zp = e2itcx27x77x21x76x33x2x29x2bx9x21
L(D)zp = e2itL(D+2i)c = e2it
x6c
L(D+2i)c = L(2i)c = 8ic = 1
x9x6cc =?i8,x24x17x2bx76x33x2x44x9x27
xp = Re
i8e2it
= sin(2t)8,
x73,x1ax41x9x4dx59x25xax8x2fx23x3ax3bx39x6ax5x76x33x2x44x9xd,x4fx17x79x6x2ax77x6dx3e
x2x59x6xax8x2fx23x3ax3bx39x6ax5x76x33x44x9x2x76x7,xexfx71x77:
x62 n x66x8x2fx23x3ax3bx39x6ax5x76x33x71x72x50x7dx76x33x36x64,Dx = Ax + f(t),x73
(ID? A)x = f(t),x59x44x31x3fxdx64 det(?I? A) = L(?),x7ax59x5a n xax8x2f
x23x3ax3bx39x6ax5x76x33 L(D)y = f(t) x2x44x9 yp,x8x2bx76x33x2x44x9x27 xp(t) =
(ID?A)?yp(t),x58x49(ID?A)? x27(ID?A)x2xax34x6bx25.
x6f:
(ID?A)xp = (ID?A)(ID?A)?yp(t) = L(D)yp(t) = f(t):
51
x73Dxp = Axp +f(t).
x6cx9x39x19x13x14x60xbx19
x6x64x39x3ax3bx25xax76x33x69x20x2fx23x3ax3bx76x33x17x2ax79x27x7bx2x9x7,x3dx52x7x52x53x58
xax23x4ax9x6x2ax2x10x2bx8; x27x27x52x53x76x33x2xax23x73x4fx52x53x9x48x2x4fx30; x44x9x3x5c
x1axax3ax3bx20x2fx23x76x33,x40x50x4cx5ax58x3xdx6ax5x76x33x2x6x2ax44x9,x8x4fx62x30x52x53x6
xa,x57x55x59x5ax3xdx6ax5x76x33x2x27x9,x67xdx59x7bx8x23x20x6dx7x55x6cx4dx2bx76x33x2x27x9.
1,n x19x13x14x8x32xcx33 x;x0;￠￠￠ ;x(k?1) x60x4x1d x28x29x76x33x36x27
F(t;x(k);￠￠￠ ;x(n)) = 0,(2:4:14)
x40x26 x(k) = y,x8(2.4.14)x52x27x63x5c y x2 (n?k) xax76x33
F(t;y;￠￠￠ ;y(n?k)) = 0,(2:4:15)
x5cx9x40x59x5a(2.4.15)x2x27x9
y = ’(t;c1;￠￠￠ ;cn?k);
x8x7x32 k x5x40x75x73x6c(2.4.14)x2x27x9
x =?(t;c1;￠￠￠ ;cn);
x58x49 c1;￠￠￠ ;cn x27x14x74x8x23.
x76 2.4.12 x59x9x76x33 tx(5)?x(4) = 0.
x2ax73x4 x26 x(4) = y,x8x2bx76x33x72x27 Euler x76x33 ty0?y = 0,x11x63x9x6c
y = ct,x73x(4) = ct,x5cx9x7x6bx5x40x75x73x6cx2bx76x33x2x27x9x27 x = c1t5 + c2t3 +
c3t2 +c4t+c5,x58x49 cj;j = 1;:::;5,x27x14x74x8x23.
x2ax73x17 x45x46x66x76x33x10x25x24x9Eulerx76x33t5x(5)?t4x(4) = 0,x44x31x76x33x9
(1)(2)(3)(5) = 0,x57x55x4x6cx66x38x27x9.
2,x13x14x8x32xcx33x5fx5cx5d t x60x4x1d x28x29x76x33x27
F(x;x0;￠￠￠ ;x(n)) = 0,(2:4:16)
x3x5cx28x2ax76x33,x40x17 x x27x25x20x26,x26 x0 = y,x8x4fx62x30x52x53x6xa,x45x46x66,x59x7b
d
dt =
dx
dt
d
dx = y
d
dx
x73x4f,x28x29 x00 = dydt = ydydx,x000 = dx
00
dt = y
d(x00)
dx = y(
dy
dx)
2 +y2d2y
dx2,￠￠￠,x59
x7bx23x41x75x22x7x4fx17xdx7f,x(k) x4fx7b y; dydx;￠￠￠ ; d
k?1y
dxk?1 x3bx72xb,x62x28x72x72x3ex64x64x65(2.4.16)x4fx6c
G(x;y; dydx;￠￠￠ ; d
n?1y
dxn?1) = 0;
x57x55x6a(2.4.16)x52x53x4dx6xa.
52
x76 2.4.13 x59x9x76x33 xx00 +(x0)2 = 0.
x2a x26 x0 = y,x8 x00 = ydydx,x5cx9x2bx76x33x72x27 xydydx +y2 = 0,x73 y = 0
x69xdydx =?y;x57xdx64x59x6c y = c=x,x73 xdx = cdt,x12x79 x2 = c1t+c2,x30x3a
x7ex4dx11 y = 0 x6cx4dx2x9 x = c,x27x63x2bx76x33x2x27x9x24x9 x2 = c1t+c2.
3,x3x5cx1axax20x2fx23x3ax3bx6ax5x76x33
x00 +p(t)x0 +q(t)x = 0; (2:4:17)
x71x13x7bx4ex30x2x6x2ax9 x = ’(t),x8x4fx59x5ax30x2x27x9,x27x63x1a(2.4.17)x49x2x7fx7b
x22x23x2x7cx7d x = ’(t)R y(t)dt,x73x4fx62(2.4.17)x20x72x63x5c y(t) x2x3ax3bx6ax5x76
x33
’(t)dydt +[2’0(t)+p(t)’(t)]y = 0;
x11x63x9x6c
y(t) = c2 exp(?
Z
([2’0(t)=’(t)+p(t)]dt = c2’?2(t)exp(?
Z t
t0
p(u)du);
x58x49 c2 x27x14x74x8x23,x5cx9(2.4.17)x2x27x9x27
x = ’(t)
c1 +c2
Z t
t0
’?2(s)exp(?
Z s
t0
p(u)du)ds
,(2:4:18)
x27x27’(t) x4e ’(t)Rtt0 ’?2(s)exp(?Rst0 p(u)du)dsx52x1ax1bx9x8x23,x24x17x30x7dx3a
x3bx4bx63,x73x74xcx1a(2.4.18)x49x2x36x40x75x4fx17x5ex27x1bx36x40x75x3cx30x59x1bx36x40x75x29x14
x16x6x2ax2bx22x23x73x4f.
x762.4.14 x7ax7b x = sintt x9x76x33 x00+ 2tx0+x = 0 x2x9,x2dx59x58x27x9.
x2a x28x28 p(t) = 2=t,x5cx9x11(2.4.18)x73x6c
x = sintt
c1 +c2
Z t2
sin2 te
R 2tdtdt
= sintt
c1 +c2
Z dt
sin2 t
= 1t(c1 sint?c2 cost);
x58x49 c1;c2 x27x14x74x8x23.
53
x41x45x43 x46x33x2dx47x46x33x48x49
x23x14x7bx4e,x50x59x5ax27x9x2x8x74x75x76x33x9x8x27x79x6fx2,x20x1bx3fx24x9x29x29x78x34
x10x2x31x72x76x33xbx1,x23x26x39x3ax3bx76x33x2x59x9x17x0x6ax26x27,x2fx35x1bx66x7bx7x8x22x23
x2x40x75x3bx72xb,x27x63x24x47x5ax53x54x57x74x75x76x33x2xex24x3bx26x27x9x2x3bx43,x69x1dx26x27
x11x74x75x76x33x24x4ex36x53x3ax2x75x64x4cx4dxcx58x20x55x3bx43.x28x24x9x31x30xdxex2fx31x5fx64x7
x4dx30x31x5fx64x7x11 H,Poincar′e x1f A,M,Liapunov x47x5ax2x74x75x76x33x36x3bx39
x10x1fx5ax36x3bx39x10.
x3.1 x21x22x23x24
1,x42x11x22x15 x6dx6x64x76x33x0 = f(t;x)),x49x22x23 f x49x1bx60x7e t x29,x73
x0 = f(x); x 2 Rn (3:1:1)
x8x60x30x27x7dx32x22x15x69x16x7dx22x15x69x5fx0x22x15,x58x49 f(x) x1a Rn x2x28x5dx30 G x49
x6ex6f,x30xcx6f (3.1.1) x60x2ax40x42x4ax35x1fx4x17x7d (x71 Lipschitz x13x6bx8)x30x9x2x36
x7x5dx13x17x27 R ;x5cx9x3? 2 G,(3.1.1) x24x79x2cx6x2x6x2ax9 x = ’(t;?) x37x38
’(0;?) =?,x57x1bx15x66x3bx1d,x9x = ’(t;?) x2x40x75x53x3ax9Rn+1 x35x13x2xax7b
f(t;x) 2 Rn+1jx = ’(t;?);t 2 Rg,x60x27x18x22(trajectory),x28x40x75x53x3ax1ax2c
x35x13x66x2x6ex1bfx 2 Rnjx = ’(t;?);t 2 Rgx9 G ‰ Rn x49x2x6x13x53x3a,x60x30
x27x6x13x18x19,x55x60 Rn x27x47x52x4a,x2cx35x13x2x4x60x27x47x5e,x6d t x20x21x29x24x7ex47
x5ex7x18x19x1cx5fx42,x55x60x2cx35x13x49x2x53x3a fx 2 Rnjx = ’(t;?);t? 0g,x27x1ax28
x18,xbx39x31,x60x2cx35x13x49x2x53x3afx 2 Rnjx = ’(t;?);t? 0g,x27x1bx28x18,x60x11
(3.1.1)x24x4ex36x2x4cx3fx5ex4ex2x18x0x27x6x2ax42x11x22x15,x7cx71,x3x5c n xax8x76x25 A,
x2fx5 x0 = Ax x2x24x79x5ex4ex24xex72x4dx6x2ax21x2cx2fx5.
x63x5c(3.1.1)x2x9 x = ’(t;?) x79x71x77x3bx43xc
x11x73 3.3.1 x40 ’(t;?) x9 (3.1.1) x2x9,x8x3x14x6x2x8x23 c,’(t + c;?)
x65x9(3.1.1)x2x9.
x11x5ex4ex2x36x7x73x79x11x73 3.1.2 x40 ’(t;?) x9(3.1.1)x2x9,x8 ’(t+c;?)
x3xdx2x5ex4ex4e c x4bx63,x73x3x14x74x8x23 c,’(t + c;?) x24x3xdx2x9 (3.1.1) x2x57
x6x13x5ex4e.
x1cx1d3.1.1 x3 8? 2 G,(3.1.1)x79x30x56x79x6x13x5ex4ex27x32x4?.
x11x73 3.1.3 (x36x3bx43) x3 8? 2 G x17x79
’(t2;’(t1;?)) = ’(t1 +t2;?),(3:1:2)
x16x45x66,x11x3bx431x73x7b,’(t;’(t1;?) x1f ’(t+t1;?) x17x9(3.1.1)x2x9,x2d
x6d t = 0 x29,x28x29x2ax9x5dx27 ’(t1;?),x12x11x9x2x2cx6x3bx73x7bx79’(t;’(t1;?)) ·
’(t+t1;?),x44x9x6d t = t2 x29x73x6c(3.1.2)x64,xdx45.
54
2,x65x59x5exdx2x13x7dx11.
x71x13 x0 2 G x51x6c f(x0) 6= 0,x8x60 x0 x27(3.1.1)x2x32x5e,x11f(x)x2x6e
x6fx3b,x57x1bx15x66x1d,x1ax8x4x5ax30x49x2x5ex4ex17x55x39x31x2cx2bx5cx4b.
x71x13 xe 2 G x51x6c f(xe) = 0,x8x60x2cx4 xe x27(3.1.1)x2x5cx22x4,x3x5cx5c
x22x4 xe x3bx7e,x8x23x77x26x = xe x9 (3.1.1) x2x9,x60x27x7dx32x2a,x58x40x75x53x3ax27
x5cx4bx5c t x78x2x53x3a,x55x58x5ex4ex24x9x2cx35x13xe x49x6x2ax4,x71x13x7cx7dx6fx6cx4dx2cx35
x13x66x5ex4ex24x37x38x2x76x33x3cx4fx17x6ax1 x6dx2x25x20x26x3cx6a(3.1.1)x71x72
dxj
dx1 =
fj(x)
f1(x); j = 2;3;:::;n?1
x8x4fx50 xe x9x17x66x5ex4ex76x33x2x1fx4,(x27x1ax4 xe x66x64x75x0x27x2),x12x65x79x6ax5c
x22x4x60x27x1fx4x2,x40x5cx22x4x1ax2bx4xc xe = 0,x28x29x7x30x66x60x30x27x2x9x3c x14x27x27
x = 0x9x9,x30x1bx60x7ex29x13x20x26t,x66x67x9x1fx5ex4ex1bx9x57x6x2ax3x4!)
x51x25 3.1.1 x40 xe x27(3.1.1)x2x5cx22x4,x8x3x14x15x1bx57x5cxe x2x5ex4ex1bx4f
x50x1ax79x6fx29x13x4dx3ex69x17x5c xe.
x6fx70 x32xd,x40x36x39xex10x1bx32,x8x3ax1a(3.1.1)x2x6x2ax39x36x8x9 x(t) x1fx79x6f
x2x29x2a?,x51x6cx6d t !? x29x79 x(t) ! xe;x11 x(t) x2x6ex6fx3bx73x7b x(?) = xe;
x11x63x73x50x32 xe x79x29x13x5ex4e,x28x4ex66x6cx2x3bx433.1.2x0x1,x50x48xdx45.
x1cx1d x40 (3.1.1) x2x9 ’(t) x6d t !? x29x79 ’(t) x17x5c (3.1.1) x2x5cx22x4
xe,x8x58x79? = +1x69?1.
x51x25 3.1.2 x40x1a xe 2 G x2x14x74x70x5ax30x1e,x17x3ax1ax29x13x5x30x27x14x74x23x2
x5ex4ex1e,x8 xe x58x27x5cx22x4.
x6fx70 x7bx32xdx7,x40 x0 x1bx9x5cx22x4,x8x58x3ax1ax29x2a t?,x51x6c ’(t?;x0) 6=
x0,x59 x0 x4e ’(t?;x0) x52x13x2x59x17 d(’(t?;x0);x0) = ‰ > 0,xfx6dx9x3x7
x2bx2x6ex6fx3b,x3x6x36x2 " = ‰=3 > 0,x3ax1a – > 0;– < ",x51x6cx3x6x45x37x38
d(x0;x1) < – < ‰=3 x2 x1,x17x79 d(’(t?;x0);’(t?;x1)) < " = ‰=3,x5ex73x11
x4 x0 x2 – x5ax30x1ex5ax1dx2x5ex4e,x7x29x13x5x30x27 t? x52xdx17x17x20x28x2ax5ax30,x28x4e
x50x48x49x2x51x52x0x1,x50x48x6cxd.
x7dx51,x7cx7dx60x2fx5(3.1.1)x2x5cx22x4xe x1ax20x21x2ex22x23(Liapunov)x74x7x77
x9x13x7dx60,x71x13x3 xe x2x14x6x5ax30 U,x19x3ax1ax6x2axe x2x5ax30 U1 ‰ U,x51x6cx2f
x5 (3.1.1) x2x1a t = 0 x29x2dx5c U1 x1ex2x5ex4ex1a t > 0 x29x21x77x2dx5c U x49,x79x8x24
x60x5cx22x4xe x1bx5ax36x9x71x13x5cx22x4xe x9x5ax36x2,x5fx30x3ax1axe x2x6x2ax5ax30U0,
x51x51x6cx2fx5(3.1.1)x2x1at = 0x29x2dx5cU1 x1ex2x5ex4ex(t)x37x38
limt!1x(t) = xe;
x8x60 (3.1.1) x2x5cx22x4xe x9x24x74x13x7dx60,x5fx30x60x28x58x2 U0 x9 (3.1.1) x2x5cx22
x4x2x25x4bx26,x40 U0 = Rn,x8x60(3.1.1)x2x5cx22x4x9x18x19x24x74x13x7dx60.
x2x27 (3–3) x5cx22x4x5ax36x3bx3x4x2xdx7b,x7cx7dx3cx1ax6ex6f n x66x8x2fx23x3ax3bx2f
x5(x76x33(system) )
x0 = Ax (3:1:3)
55
x2x5cx22x4x2x5ax36x3b,x7cx7dx56x34x10A x27 n xax39x1fx4fx45x8x76x25x2x75x76,x28x29x2fx5
x2x5cx22x4x9xax6x2,x = 0,x11x2.3 x7bx4e,(3.1.3) x2x27x9x27 x = exp(At)c,x28
x28x2 c x27x14x74x2 n x66x45x8x77x26,x55x33x30x2x9x25 exp(At) x11 (2.3.6)–(2.3.7)
x6x5a,x11x33x30x2x9x25x2x36x64x6cx6dxdx7fx71x77x36x39xc
x7dx4c 3.1.1 x40 A x2x44x31x2bx17x3cx79x5ax45x4d,x8 (3.1.3) x2x5cx22x4x9x20x55x5a
x36x2;x40 A x2x44x31x2bx69x6ax79x6x2ax31x45x4d,x8(3.1.3)x2x5cx22x4x9x1bx5ax36x2;x40A
x2ax79x31x45x4dx44x31x2b,x3dx79x2x45x4dx44x31x2b,x30x3x5cx3dx6x2ax3fx23x27?n x2x2x45x4dx44x31
x2b,x76x25 AI x2x63x5dx27 nn,(x7cx71x2x45x4dx2x44x31xfx17x9x3exfx29x24x9x28
x58)x8(3.1.3)x2x5cx22x4x9x5ax36x2,x3dx1bx9x20x55x5ax36x2;x40 A x2ax79x31x45x4dx44x31x2b,
x3dx79x2x45x4dx44x31x2b,x30x69x6ax79x6x2ax3fx23x27?n x2x2x45x4dx44x31x2b,x76x25 AI
x2x63x23x5cnn,x8(3.1.3)x2x5cx22x4x9x1bx5ax36x2.
3,x27x28x2ax7ax29x18
x71x13x2fx5 (3.1.1) x2x9 ’(t);t 2 R,x9 t x2x5cx37x22x23,x73x3ax1a ! > 0,x51
x6cx3x6x45 t 2 R x79 ’(t + !) = ’(t),x8x60 ’(t) x27 (3.1.1) x2x27x28x2a,x55x60
! x27x30x2x27x28,x60x67,x36x8x9x9x17x14x6x8x23x27x58x5cx37x2x5cx37x9,x57x1bx15x66x1d,x1a
Rn+1 x35x13x49,x5cx37x9x2x40x75x53x3ax9x6x13x17xex70x31x5cx37x27x2ax59x2x2ax2bx3a,x55x58
x3xdx2x5ex4ex9x1ax2cx35x13 Rn x49x2x6x13x7fx53x3a; x1bx9x36x8x9x2x5cx37x9x24x3xdx2
x5ex4ex60x27x29x18.
x51x25 3.1.3 x1bx9x8x2bx2x6ex6fx2x5cx37x22x23x3cx79xex70x31x5cx37,x30x28x2ax5cx37x22
x23x2x5cx37x17x9xex70x31x5cx37x2x31x15x23x6b,x27x63x7fx5ex3xdx2x9x3cx79xex70x31x5cx37.
x6fx70 x26T x27x5cx37x22x23x2x24x79x31x5cx37x2x18x0,x52!0x27x18x0T x2x77x4ex41x3cx40
!0 x27x18x0T x49x2x49x4x4x3cx8!0 x24x9xex70x31x5cx37,x3cxd!0 x58x1bx9x18x0T x2x32
x4,x57x55xdx7fx4dx10x50x48.x7bx32xdx7x3cx40!0x9T x2x32x4x3cx8x3ax1ax50x21 !1,!2,￠￠￠,
!k,￠￠￠,x6d k !1x29,!k ! !0,x30!k > !0,xfx6dx22x23x2x5cx37x3bx1fx6ex6fx3bx3cx3
x5cx14x74t,x79’(t) = ’(t+!k) = limk!1’(t+!k) = ’(t+!0),x73!0 x65x9x22
x23x2x5cx37,x1bx11x5cx37x3bx3cx3x5cx14x6t 2 R,’(t) = ’(t?!0) = ’(t+(!k?!0)),
x27x63 Tk,= !k?!0 > 0 x65x9x22x23x2x5cx37x3cx30x6d k ! 1x29x3c Tk ! 0 x57x55x11
’(t)x2x5cx37x3bx17xcjt?[t=Tk]Tkj < Tk (x58x49[￠]x72xbx16x15x20x50x3ex6c
’(t) = ’(t?[t=Tk]Tk) = lim
k!1
’(t?[t=Tk]Tk) = ’(0)
x28x4e ’(t) x27x39x8x2bx22x23x0x1,x24x17x79xex70x31x5cx37!0.
x52! x9’(t)x14x6x2ax5cx37,x8’(t) = ’(t+!) = ’(t+(!?[!=!0]!0)),x27
!0 x9xex70x31x5cx37,x24x17!=!0 x9x31x15x23,xdx45
4,x18x19x60x2cx2dx67x47
x51x52x2fx5 (3.1.1) x2x5ex4ex18x0xex72x6x2ax21x2cx2fx5,x73x3dx13x5ex4ex3xdx2x15x1f
x9x17x1a R x66x79x36x7,x27x4dx26x27x15x2ax2fx5x5ex4ex2x75x64x75x76,x79x7axdx6dx3dx60x3ex6
x13x5ex4ex2x24x74x11x4a,x73x6d t ! +1 x1f?1 x29x5ex4ex2x8x6fx75x76.
x52 ‘,x = ’(t;P);t 2 R,x27x2fx5(3.1.1)x2x14x6x5ex4e;x71x13x3ax1ax29x13 t x2
x50x21ftkg,x6d k !1 x29,x79 tk !1,x51x6c
lim
k!1
’(tk;P) = Q 2 Rn; (3:1:4)
56
x8x60x4 Q x27x5ex4e ‘ x69x31x71x5e ‘+ x2 ! x2cx2dx5e,x57x58,x40x3ax1ax29x13 t x2x50x21
ftkg,x6d k ! 1 x29,tk !?1,x51x6c(3.1.4)x72x4,x8x60x4 Q x27x5ex4e ‘ x69x5a
x71x5e ‘? x2 fi x2cx2dx5e.
x5ex4e ‘ x2x24x79 ! x8x6fx4xex72x2x18x0x60x27x30x2!x2cx2dx67x47,x2ex2fx30?‘;x57
x58,‘ x2x24x79 fi x8x6fx4xex72x2x18x0x60x27x30x2 fix2cx2dx67x47,x2ex2fx30 A‘.
x7dx4c 3.1.2 x79x41x5ex4e ‘ x2x8x6fx18?‘(A‘) x9x6x2ax39x35x42x79x41x42x7fx2x6ex27
x18x0,x30?‘(A‘)x8x5cx40x4bx15x13x5ex4ex2x33x18.
x6fx70 x11x5c ‘ x2x79x41x3b,x12?‘(A‘) x9x6x2ax39x35x79x41x18,x2dx11x8x6fx18x0x2
x36x7,x4fx53x54x70x5a?‘(A‘) x27x7fx18,x1bxdx7fx58x6ex27x3b.
x32xd,x40?‘ x1bx6ex27,x8x58x3ax39x35x2x42x2bx1bx2cx26x2x79x41x7fx18?1‘ x1f?2‘ x51
x6c?‘
=?1‘ [?2‘,x30x30x7dx52x13x2x59x17 d(?1‘;?2‘) = ‰ > 0.
x52x3xdx5cx5ex4e ‘ x2x9x27 x = ’(t;p),x5cx9x3ax1ax23x21 0 < t1 < t2 < ￠￠￠ x51
x6c d(’(t2k+1);?1‘) ! 0,d(’(t2k;p);?2‘) ! 0,x6d k ! +1,x73 tk ! +1.
x57x55x3ax1ax25x67x23 N,x51x6cx3x3dx2a k > N x17x79 t?k,t2k < t?k < t2k+1,x30x37x38
d(’(t?k;p);?i‘)? ‰=4;i = 1;2;k > N:
x11 ‘ x2x79x41x3bx73x7bx50x21 ’(t?k;p)(k > N) x58x3ax1ax8x6fx4 x0,x60x67 x0 2?‘,
x30 d(x0;?i‘)? ‰=4;i = 1;2;x57x55 x0ˉ2?1‘ [?2‘,x28x24x4e?‘ =?1‘ [?2‘ x0
x1,x12?‘ xdx27x6ex27x18.
x52 q 2?‘ q1 = ’(t1;q) x27x32q x4x2x5ex4ex66x2x14x6x4x11x5c q x9 ’(t;p)
x2 ! x8x6fx4,x24x17x3ax1ax50x21 ftkg;k = 1;2;:::;tk ! +1,x51x6c ’(tk;p) ! q,
x6d k ! +1,x11x9x2x36x3bx433x4ex9x3x7x2bx2x6ex6fx3b,x6d tk ! +1 x29x79
’(t1 +tk;p) = ’(t1;’(tk;p)) ! ’(t1;q) = q1;
x5ex73 ˉtk,= t1 +tk ! +1x30’(ˉtk;p) ! q1,x28x7ex7f Q1 x65x9x5ex4e ’(t;p) x2
! x8x6fx4,x11 q x1f t1 x2x14x74x3bx3c?‘ x8x5cx40x4bx15x13x5ex4ex2x33x18,x57x39x4fxd A‘
x2x75x76,x36x39xdx45.
x1cx1d 1 x40 ‘ x9x2fx5(3.1.1)x2x14x6x7fx5e,x8?‘ = A‘ = ‘.
x1cx1d 2 x40x79x41x5ex4e ‘ x2 !(fi) x8x6fx18x0?‘(A‘) x9x56x11x6x4 Q x24x72
x2,x73?‘(A‘) = fQg,x8 Q x58x27x5cx22x4,x5fx30x6d t ! +1(?1) x29x5ex4e ‘ x17
x42x5cx28x2ax5cx22x4,x32x52,x40x6d t ! +1(?1) x29x5ex4e ‘ x17x42x5cx6x2ax4 Q 2 G,
x8 Q x58x27x5cx22x4,x30x79?‘(A‘) = fQg.
x71x13x5ex4eLx66x2x3dx6x4x17x9x5ex4e ‘ x2 !(fi) x8x6fx4,x8x60 L x27 ‘ x2
!(fi)x2cx2dx18x19.
57
x3.2 x35x36x2ex37x38x55x31x1dx39
x10x3x79x7ax33x1dx34x10x2cx5cx6cx66x3ax3bx2fx5x5cx22x4x2x75xb,x58x5x37x38x79x41x5ex4ex8x6f
x18x0x2xex24,xexdxexfx79x63x2cx5cx6cx66x8x6fx34x2x6x72x2x10x7bx29.
x6x42x17x34x32x22x23x22x11x22x15x65x59x5ex60x12x35
x7cx7dx3cx1ax6ex6ax1ax66x3ax3bx2fx5
x0 = Ax (3:2:1)
x58x49 A x27x1axax8x45x25,x30 detA 6= 0 x11x63x73x50 (3.2.1) x79x2cx6x2x5cx22x4 x =
(0;0)T.
x27x34x10x5cx22x4x54x55x5ex4ex2x3bx43x3cx7cx7dx27x32x2cx35x13x32x33x2x3ax3bx20x7dx3bx6dx21x76
x33.
x52x6bx25 A x2x29x2ax44x31xfx27?1(6= 0),?2(6= 0),x30x52x30x7dx9x1bx2cx8x2x45x23x3c
x8x3ax1ax29x2ax3ax3bx4bx63x2x67x45x44x31x77x26 v1,v2,x73x51x6c Av1 =?1v1,Av2 =
2v2,x62x28x29x2ax3ax3bx4bx63x2x67x45x44x31x77x26xex72x6bx25 T = (v1;v2),x5fx59x3x57x25
J = diag(?1;?2)x3cx8x4fx71x72x6bx25x36x64AT = TJ,x5cx9x4fx2x3ax3bx20x7d
x = Ty
x5cx9x76x33x21x27
x0 = Ty0 = Ax = ATy0
x73x6cx76x33
y0 = T?1ATy = Jy0
x59y = (y1;y2)T,x8x27 ‰
y01 =?1y1;
y02 =?2y2:
x28xex76x33x1at = 0x29x32x4(y10;y20)T x2x9x27
y1(t) = y10e?1t; y2(t) = y20e?2t; (3:2:2)
x4fx50x31x5ay1 x71x78x3cx31x5ay2 x71x78x9x6bx13x5ex3ax3cx78x79x2bx4x9x5ex3ax3cx6bx4dx28x44x45x2
x56x13x5ex3ax6x3cx58x41x2x5ex3ax4fx17x57(3.2.2)x49x3ax4ax20x26tx6cx4d,x5ex3ax2x76x33x27xc
y2 = y20(y1=y10)?2=?1; y10 6= 0; y20 6= 0:
x11x5c?2=?1 6= 1,x4fx50x28x72x5ex3ax17x2cx39x5cx5ex3a
y2 = §jy1j?2=?1
x24x17x1ax5cx2cxax29x3cx56x6fx5cx6x13x5ex3ax3cx67xdxfx6dx3x60x3bx1fx2cx39x3bx4fx5cx5ax58x41x5e
x3a,x5ex3ax2x36x4cx11x1ax2b?2=?1 x4dx36,x5ex3ax2x39x42x11x44x31x2bx2x31x5ax3bx4dx36,x17x77
x75x1bx69x75x76x3bx34x10.
x75x761,x29x44x31x2bx27x57x2dx3dx1bx2cx8x3cx1bx5bx52?2=?1 > 1,(x27x7cx7dx19x4fx17x77x72
x44x31x2bx2x0x2dx51x30x72x4)x3cx6bx4dx44x45x2x56x13x5ex3ax6x3cx58x41x2x5ex3ax9(xcx7x2)x4bxb
58
x3a,x6bx29x13x5ex3ax6,x58x30x5ex3ax17x4ey1 x78x2cx45x5cx5cx22x4,x58x5ax30x1ex3cx79x28x58x3bx4d
x5ex3ax2x5cx22x4x60x27x36x5e.
x6dx29x2ax44x31xfx5dx27x5ax23x29,x66x38xex4x27x20x55x5ax36x2,x7cx7dx60x28x58x2xex4x27x24
x74x13x7dx36x5e.
x6dx29x2ax44x31xfx5dx27x31x45x23x29x3cx66x38xex4x9x1bx5ax36x2,x57x55x58x3xdx2x5cx22x4
x60x27x32x13x7dx36x5e.
x75x762,x29x44x31x2bx4fx2dx3cx1bx5bx52?2=?11,x6bx4dx44x45x2x56xfx5ex4ex6x3cx58
x41x5ex4ex9x14xcx7x2x3ex4dx53x3ax3cx5ex3ax2x75x64xbx39x5cx3ax3bx6cx2x8x2dx3a,x12x60x28x69x5c
x22x4x27x3cx5e,x3bx4x9x1bx5ax36x2.
x75x763,A =?I,x1at = 0x29x7x32x4x0 x2x9x27
x = exp(At)x0 = exp(?It)x0 = exp(?t)x0
x6bx4dx5cx22x4x6,x58x41x2x5ex4ex27x57x5cx22x4x5ax1d,x76x42x27x0 (x1bx3ax3bx5cx22x4)x2x63x3a:
x28x58x2x5cx22x4x60x27x39x36x5ex65x60x27x34x36xex4,x1fxex4x2x5ax36x3bx11? x2x2cx2d
x4dx36xc? < 0 x29x27x20x55x5ax36x1fxex4,? > 0 x27x1bx5ax36x1fxex4.
x75x764,?1 =?2 =?,x30A 6=?I x29x3cx28x29x3ax1axcx7x45x44x31x77x26x61v1,v2x3c
x16T = (v1;v2),x51x6c(AI)v2 =,v1,x8x79AT = TJ,x58x49J =
1
0?
.
x5cx9x4fx2x3ax3bx20x7d
x = Ty
x5cx9x76x33x21x27
x0 = Ty0 = Ax = ATy0
x73x6cx76x33
y0 = T?1ATy = Jy0
(3.2.1)x20x72 ‰
y01 =?y1 +y2;
y02 =?y2;
x58x9x27
y1(t) = (c1t+c2)exp(?t); y2(t) = c1 exp(?t); (3:2:3)
x58x49 c1;c2 x27x14x74x8x23,x6bx4dx65x13x44x45x2x5ex4ex14x2bx4x1fx31x5ay1 x71x78x3ex6x3cx3ax4a
(3.2.3)x49x2t,x6cx58x41x2x5ex3ax76x33:
y1 = y2 (ln(y2=c))=?
x30x1fx17x77x5ex4ex2cx39:
y1 = y2 (ln(§y2))=?
x6bx4dx5cx22x4x6,x24x79x5ex3ax1ax5cx22x4x5x17x4ey1 x78x2cx45x5cx2bx4,x3cx79x28x69x3bx43x2x5c
x22x4x60x27x3ex61x36x5e,x6d? < 0 x29x3cx60x5cx22x4x27x24x74x13x7dx60x3ex61x36x5e,x6d? > 0
x29,x5cx22x4x9x32x13x7dx60x3ex61x36x5e.
x75x76 5,?12 = fi§ifl,fl 6= 0 x27x6x3x5fx61x60xf,x27x4ex36x4fx50x3cx6d fi 6= 0
x29x3cx7cx7dx4fx17x77x16x58x49x2x6x2ax44x31x2b? = fi + ifl,x51x6c?fi=fl =,! > 0; x6d
59
fi = 0 x29x3cx4fx17x77x16x58x49x2x6x2ax44x31x2b? = ifl,x51x6c fl < 0,x52x3xdx2x44x31x77
x26x27v = v1 +iv2,x58x49v1,v2 x58x17x27x39x2x45x77x26x3cx5cx9
A(v1 +iv2) = (fi+ifl)(v1 +iv2):
x26x58x45x4dx4ex60x4dx75x9x2cx8x3cx6c
‰ Av
1 = fiv1?flv2
Av2 = flv1 +fiv2;
x73AT = TJ.
x58x49J x27x45x6bx25J =
fi?fl
fl fi
,x17x77xdx7fv1 x1fv2 x9x3ax3bx4bx63x2:
xdx3cx7bx32xdx7,x40x1bx67,x8x3ax1ax39x2x8x23cx51v2 = cv1,x5cx9Av = Av1(1+
ci) = (fi + ifl)v1(1 + ci),x73 Av1 = (fi + ifl)v1,x6cx0x1(x27x27x8x64x0x68x9x45x77
x26x3cx55x8x64x67x68x9x60x77x26),xdx45.
x27x63 T x9x39x1fx4fx2x6bx25,x5cx9x57x58x4fx2x3ax3bx20x7dx = Ty x5cx9x76x33x21x27
x0 = Ty0 = Ax = ATy0,x73x6cx33x30x2x45x76x33y0 = T?1ATy = Jy0,x73
y01 = fiy1 +fly2; y02 =?fly1 +fiy2; (3:2:4)
x59x7bx8x32x33x20x7d,x73x26 y1 = rcos ; y2 = rsin,x5fx73x74x4d r2=2 =
(y21 +y22)=2,= arctan(y2=y1),x3tx59x73x75x9x6c
rr0 = y1y01 +y2y02; 0 =?y1y02?y2y01￠=r2:
x62(3.2.4)x64x65x6c
rr0 = fir2; 0 =?fl:
x11x63x73x6c(3.2.4)x2x8x32x33x36x64x27x9
r(t) = r0 exp(fit); (t) =?flt+ 0; (3:2:4)?
x58x49 r0? 0x1f 0 x5dx27x14x74x8x23,x3ax4ax20x26t,x6cx5ex4ex76x33:
r = cexp(? fi=fl); c? 0; (3:2:5)
x6bx4dx44x45x2x5ex4ex2bx4x6,x58x41x5ex4ex17x2cx39x5cx53x3a
r = exp(? fi=fl); c? 0;
x57 (3.2.5) x73x4fx1dx5a,x6bx5cx22x4x6,x6d fi 6= 0 x29,(3.2.4) x2x5ex4ex9x3x23x2a
x2bx3a,x57 (3.24)* x4fx50,x6d fl > 0 x29x76x29x59x3fx2b,x6d fl < 0 x29x58x29x59x3fx2b,x6d
fi > 0x29,x34x4dtx4x23x3cx5ex4ex66x2x4x3fx2bx31x38x17x5cx22x4,x6dfi < 0x29x3cx34x4dtx4
x23x3cx5ex4ex66x2x4x3fx2bx31x17x5cx5cx22x4,x28x58x2x5cx22x4x60x27x40x5e,x6dfi < 0x29x60x27
x20x55x5ax36x44x4;x6d fi > 0 x29x60x27x1bx5ax36x44x4.
x6d fi = 0 x29,x5cx9x11 (3.2.5) x73x50 r x27x8x23,x27x63x5ex3ax9x17x5cx22x4x27x49x1b
x2x5bx54.x28x29x5cx22x4x24x60x27x8x41x5e,x49x1bx9x5ax36x2x3dx1bx9x20x55x5ax36x2.
60
x42x66x24x38,x4fx6cx71x77x36x39xc
x7dx4c 3.2.1 x71x13x1ax66x8x2fx23x45x3ax3bx2fx5(3.2.1)x2x2fx23x6bx25x9x39x1fx4fx2,
x8x2fx5x2x5cx22x4x62xfx6dx44x31x76x33x2xfx2x3bx43x55x75x9x3cx79x77x21x2x1bx57x44x3bxc
(i) x71x13x44x31xf?1 6=?2 x27x45xf,x8x6d?1,?2 > 0 x29x5cx22x4x27xex4,x30
x6d?1 < 0 x29xex4x9x20x55x5ax36x2,x3dx6d?1 > 0 x29,x5cx22x4x27x1bx5ax36x2; x6d
1;?2 < 0 x29,x5cx22x4x27x3bx4,x3bx4x9x1bx5ax36x2.
(ii) x71x13x44x31xf? x27x3fxf,x8x6dx6bx25 A x1bx9x23x26x6bx25,A 6=?I x29,x5cx22x4
x27x43x21xex4,x1bx67,x5cx22x4x27x1fxex4,x2dx6d? < 0 x29,x28x29xbxex4x5dx27x20x55x5a
x36x2;x6d? > 0 x29,x5cx22x4x1bx5ax36.
(iii)x71x13x44x31xfx27x5fx61x60xf,x73?fi§ifl,fl 6= 0,x8x6d fi 6= 0 x29x5cx22x4x27
x44x4,x30x6d fi < 0 x29x63x44x4x9x20x55x5ax36x2,x55 fi > 0 x29,x5cx22x4x9x1bx5ax36x2;
x6d fi = 0 x29,x5cx22x4x27x49x1b,x5cx22x4x9x5ax36x2x3dx1bx9x20x55x5ax36x2.
x76 3.2.1 x61x36x2fx5 x0 = x+2y; y0 =?2x+5y x5cx22x4xbx1x1fx5ax36x3b.
x2a x44x31x76x33?2?6?+9 = 0x79x1ax3fxf? = 3 > 0,x55x2fx23x6bx25x1bx9x23x26
x6bx25,x27x63x5cx22x4x27x1bx5ax36x43x21xex4.
x1ax42x17x34x22x15x18x19x2cx2dx67x47x60x36x48x7ax2cx2dx47
x7cx7dx68x4dx6x64x2x1ax66x36x8x2fx5
x0 = P(x;y); y0 = Q(x;y); (3:2:6)
x58x49 P,Q x1ax2cx5cx6cx2x5dx30 G x66x6ex6fx4fx74,x57x55x31xd (3.2.6) x2x9x11x7x2bx24
x2cx6x4ex36,x51x52(3.2.6)x24x79x2x5ex4ex24x72x6x2ax21x2cx2fx5.
x40 ‘ x27 (3.2.6) x2x6x13x79x41x5ex4e,x8x58?‘(A‘) x2xex24x9x77x58x78x7cx7dx1bx14
xdx7fx31x6x5ax29x13x36x39,x2x27x28x2ax47x48x2x68x1e.
x7dx4c 3.2.2 (Poincar′e-Bendixson) x40x5ex4e ‘ x79x41,x30?‘(A‘) x1bx7ex79
x5cx22x4,x8x69x1d ‘ x10x25x24x9x6x13x7fx5e,x69x1dx30x2?‘(A‘) x9x6x13x7fx5e L,x30x6d
t ! +1(?1) x29,‘ x3fx2bx7cx55x5c L.
x7dx4c 3.2.3 x40x5ex4e ‘ x79x41,x30?‘(A‘) xex3fx56x3ax7ex79x6fx2ax5cx22x4;x8x77x21
x65x69x75x36x52x6x72x4xc
(i)?‘(A‘) x11x2cx6x2x6x2ax5cx22x4 Q xex72;x28x29x6d t ! +1(?1) x29 ‘x17
x5c Q;
(ii)?‘(A‘) x11x6x13x7fx5e L x24x72;x28x29x6d t ! +1(?1) x29,‘ x3fx2bx31x7cx55
x5c L;
(iii)?‘ (A‘) x9x11x79x6fx2ax5cx22x4x1fx6x72x8x6fx5ex4ex24x72,x28x29x6d t ! +1(?1)
x29,x28x72x8x6fx5ex4ex17x71x25x75x9x17x5cx28x72x5cx22x4x6dx49x2x6x2a.
x63x5cx7fx5ex7cx7dx78x79x71x77x36x39xc
x7dx4c 3.2.4 x1ax14x15x6x2ax7fx5e ‘ x2x1ex30x1e,x17x69x6ax7ex79x6x2ax5cx22x4.
x36x7,x49x4x2x7fx5ex60x27x2cx2dx47( limit circle),(x49x4x2x74x3ax9x3ax1ax7fx5ex2
x6x2ax5ax30,x1ax28x5ax30x1ex2ax79x58x41x7fx5e)
61
x63x5cx1ax66x2fx5x2x7fx5e ‘ x2x5ax30x2x5ex4ex75x64x1fx39x42x3cx79x71x77x1bx69x75x76xc
(a)x35x13x7dx47 x57x7fx5e ‘ x2x28x6x6x4ex34x36x5ax30x1ex2x5ex4ex17x9x39x7fx5ex3cx30x17
x17 ‘ x27x30x7dx2x37x5f ! x8x6fx5ex4e,x73t ! +1x29x3cx3fx2bx17x42x7fx5e‘.
(b)x1ax13x7dx47 x57x7fx5e ‘ x2x28x6x1ex4ex34x36x5ax30x1ex2x5ex4ex17x9x39x7fx5ex3cx30x17
x17 ‘ x27x30x7dx2x37x5f ! x8x6fx5ex4e,x73t ! +1x29x3cx3fx2bx17x42x7fx5e‘.
(c) x35x32x13x7dx47 x57x7fx5e ‘ x2x28x6x6x4ex34x36x5ax30x5ex4ex2x5ex4ex17x9x39x7fx5ex3c
x30x17x17 ‘ x27x30x7dx2x37x5f fi x8x6fx5ex4e,x73t !?1x29x3cx3fx2bx17x42x7fx5e‘.
(d)x1ax32x13x7dx47 x57x7fx5e ‘ x2x28x6x1ex4ex34x36x5ax30x5ex4ex2x5ex4ex17x9x39x7fx5ex3c
x30x17x17 ‘ x27x30x7dx2x37x5f fi x8x6fx5ex4e,x73t !?1x29x3cx3fx2bx17x42x7fx5e‘.
(e)x35x27x28x47 x7fx5e ‘ x79x28x6x6x4ex34x36x5ax30x27x7fx5ex24x1fx37.
(f)x1ax27x28x47 x7fx5e ‘ x79x28x6x1ex4ex34x36x5ax30x27x7fx5ex24x1fx37.
(g)x2x47x47 x1a ‘ x2x14x74x70x5ax30x1ex31x79x7fx5ex65x79x39x7fx2x5ex4e.
x7cx7dx60x1ex6x17x5ax36x2x34x27x5ax36x2x8x6fx34x3cx60x1ex6x17x1bx5ax36x2x34x27x1bx5ax36x2
x8x6fx34x3cx60x6x4ex5ax36x4ax6x4ex1bx5ax36x2x34x60x27x71x5ax36x8x6fx34.
x76 3.2.2 x6x36x1ax66x2fx5
x0 = y +x(1?x2?y2); y0 =?x+y(1?x2?y2); (3:2:7)
x2dx34x10x58x8x6fx34x2x3ax1ax3b,x40x3ax1ax4cx5ax30x5fx30x4ex36x58x5ax36x3b.
x2a x27x4dx26x27x28x2ax2fx5x8x6fx34x2x3ax1ax3b,x6ex6fx2cx35x13x2x53x57x32x33 (x;y) x4d
x8x32x33(r; )x2x32x33x20x7d,x = rcos ; y = rsin ;x4fx50
r2 = x2 +y2; = arctan(y=x):
x3tx59x73x6cx3c
2rr0 = 2(xx0 +yy0); 0 = (xy0?yx0)=(x2 +y2)
x73
r0 = (xx0 +yy0)=r; 0 = (xy0?yx0)=r2
x62(3.2.7)x64x65x6cx8x32x33x77x2x74x75x76x33xc
r0 = r(1?r2); 0 =?1
x7cx7dx7bx4ex3cx6bx4dx2bx4x17x6x3cx53x57x32x33x66x2x4x1fx8x32x33x66x2x4x9x6x6x3xdx2x3cx44
x9x31x3cx8x32x33x76x33x49x2x5cx22x4x3xdx5cx53x57x32x33x49x76x33x2x5cx22x4x3cx53x57x32x33x49x2
x8x4x3xdx5cx8x32x33x49x2x8x4,(x3dx9x6fx73x74x3cx2bx4x9x53x57x32x33x49x2x5cx22x4x3dx1b
x9x8x32x33x36x64x76x33x2x5cx22x4),x27x63x3cx57x8x32x33x36x64x2x76x33x4fx50x3cx6bx2bx4x17x6x3c
x2ax79x58x41x5cx22x4,x30x79x9
r = 1; =?t+c
x1ax2bx32x33x77x9x6x2axex70x31x5cx37x272… x2x5cx37x9.
x = cos = cos(?t+c); y = sin = sin(?t+c)
62
x3xdx2x5ex4ex9x7fx5ex3cx9x5bx1bx1ax2bx4x2x3ex2dx5b x2 + y2 = 1,x3cx1ax4fx17xdx7fx28x2a
x7fx5ex9x76x33x2cx6x2x7fx5e,x7bx32xdx7,x40x79x4ax6x2ax7fx5e,x11x5cx1bx57x2x5ex4ex1bx2cx26,
x28x2ax7fx5ex56x4fx50x79x65x69x75x76,1,x1ax5b r = 1 x6x3cx3dx1bx3ax4ax5b r = 1; 2,x1ax5b
r = 1x6x3cx3ax4ax5br = 1; 3,x1ax5br = 1x1e.
x79x7ax3cx19x6x69x75x76x9x1bx4fx50x2x3cx27x27xfx6dx36x397x3cx28x7fx5ex62x3ax4ax6x5cx22x4x3c
x28x4ex76x33x56x79x6x2ax5cx22x4x0x1x9
x1bx1dx19x1ax69x75x76x3cx28x29x1a r = 1 x6x2x7fx5e r > 1,x57x76x33x1dx4dx1ax28x7fx5ex66
r0 < 0,x27x63x7cx7dx71x13x1fx4dx7fx5ex2x6x5cx59x40x75x6cx71x77x0x1.
0 = r(2…)?r(0) =
Z 2…
0
r0dt < 0:
x27x63x19x1ax69x75x76x65x9x1bx4fx50x2.
x1ax19x65x69x75x76x2x7fx5ex66r0 > 0x3cx57x39x4fxdx1bx3ax1ax58x41x7fx5e.
x27x63x3cx10x7cx2x2fx5x3ax1ax2cx6x2x7fx5er = 1,x9x8x6fx34,x55x30x57x34r = 1x2x1e
x6x4er0 x2x31x5ax3bx4fx7br = 1x9x1bx5ax36x2x8x6fx34.
x3x5cx6x64x2x1ax66x39x3ax3bx2fx5x3cx5fx1bx9x19x4fx17x7bx8x32x33x20x7dx3bx59x5ax8x6fx34
x2x3cx27x63x77x6cxexfx6x2ax61x1cx34x2x3ax1ax3bx2x36x39
x7dx4c 3.2.5 (x47x26x7dx4c) x3x5cx2fx5(3.2.6),x40x1ax30 G x1ex3ax1ax11x29x13x6dx3e
x7fx53x3a Γ1,Γ2 x24x4ax72x2x7fx34x30 D x30x37x38x71x77x13x6bxc
(i) Γ1 x1a Γ2 x2x1ex30;
(ii) D x66x1bx7ex79x5cx22x4;
(iii) x24x79x4e D x2x1ex68x41 Γ1 x1fx6x68x41 Γ2 x2cx26x2x5ex4ex17x1a t x4x14 (x21
x70) x29x57 D x2x6x4dx2bx65x30x2x1ex4d; x8x1a D x1ex69x6ax3ax1ax6x2ax6x4ex5ax36x14x1bx5a
x36x3ex34x1fx6x2ax1ex4ex5ax36(x1bx5ax36)x34(x28x29x2ax34x4fx50x17x9x8x6fx34x3cx65x4fx50x3fx0x72x6
x2ax5ax36(x1bx5ax36)x8x6fx34),x30(x7d)x3ax4a D x2x1ex68x41 Γ1 x1ax58x1ex4d.
x77x6cx9x6x2ax61x1cx2fx5x1bx3ax1ax7fx5ex2x1fx75x3bx36x39
x7dx4c 3.2.6 x40x2fx5 (3.2.6) x2x67x1x22x23 P(x;y),Q(x;y) x1ax3ex6ex27x5dx30
Dx1ex6ex6fx4fx74,x30@P@x + @Q@y x1aDx1ex1bx20x2dx42x1aDx2x14x15x7dx30x49x1bx7fx27x2;x8
(3.2.6)x1bx3ax1ax4cx4dx2dx5cD x1ex2x7fx5e.
x6fx70 x32xd,x40x1a D x1ex3ax1ax7fx5e L,x = x(t); y = y(t);0? t? !,x28x28
! x27x7fx5ex2xex70x31x5cx37,x17? x59x11 L x4ax72x2x7fx5dx30,x5cx9x6d Green x37x64x1f
(3.2.6)x79
R R
(@P@x + @Q@y )dxdy = R
L
Pdy?Qdx
= R!0 (P dydt?Qdxdt)dt
= R!0 (PQ?QP)dt = 0
x28x4ex36x39x51x52x0x1,x36x39xdx45.
63
x3.3 x2cx2dx3ax30x31x2ex37x51x52x53x38x3bx55x31
x7cx7dx3cx1ax68x4dx6x64x36x8x2fx5 (3.1.1) (x73 x0 = f(x)) x2x5cx22x4x2x5ax36x3b.
x40 xe 2 G x51x6c f(xe) = 0,x8x1ax2cx35x13 Rn x49x4 xe x24x9(3.1.1)x2x5cx22x4,
x55 x = xe x27x30x2x36x8x9.
x6x42x4fx22x11x74x78x50x7dx23x22x11x13x14x65x59x5ex60x13x7dx11
x40x22x23x77x26 f(x) x1a xe x5x6ex6fx4fx74,x8x6dTaylorx37x64x4fx6a f(x) x71x72
f(x) = @f@x(xe)(x?xe)+R(x;xe); (3:3:1)
x58x49 n xax76x25 @f(xe)@x x9 n x66x22x23x77x26 f(x) x3 n x66x77x26 x x2 Jacobi
x6bx25x1a xe x5x2x2b,x55 n x66x22x23x77x26 R(x;xe),= f(x)? @f(xe)@x (x?xe),x11
(3.3.1)x73x50x30x37x38
R(xe;xe) = 0; lim
jxj!xe
jR(x;xe)j=jx?xej = 0,(3:3:2)
x40x59 @f(xe)@x = A,x8(3.3.1)x4fx71x72
x0 = Ax+R(x;xe); (3:3:3)
x28x28 R(x;xe) x37x38(3.3.2),x27x63x6djxjx3cx70x29,jR(x;xe)j x9jx?xejx2x25xa
x4bx6fx70x26,x5cx9x6dx7cx7dx1ax26x27(3.3.3)x5cx22x4x2x5ax36x3bx29,x9x79x4fx17x7bx26x27x3ax3b
x76x33(3.1.3) (x73 x0 = Ax) x5cx22x4x2x5ax36x3bx3bx64x7cx73x78x7cx7dx60x76x33 x0 = Ax
x27 (3.3.3) x2x3ax3bx55x39x76x33x69x6x5x55x39x76x33,x55 x0 = Ax x2x5cx22x4x2x5ax36x3b
x47x48,x7cx7dx7ax1ax3.1x49x2x36x393.1.1x9x4dx4d.
x7dx4c 3.3.1 x1ax13x6b(3.3.2)x77,x40x44x31x76x33 det(?I?A) = 0 x2xfx5dx3cx79
x5ax45x4d,x8(3.3.3)x2x5cx22x4x9x20x55x5ax36x2;x55x6dx30x2xfx69x6ax79x6x2ax3cx79x31x45x4d
x29,(3.3.3)x2x5cx22x4x9x1bx5ax36x2.
xdx7bx36x393.3.1 x3bx61x1cx76x33 (3.3.3) x5cx22x4x5ax36x3bx78x79x6x2ax55x4f,x28x24x9x6d
x2fx5x2x66x23x77x23x29,x44x31x3fxdx64x2x5x23x77x23x3cx3cx4fx7bx4ex30x2x44x31xfx9x79x4cx3cx79
x5ax45x4d.x27x63x7cx7dx79x71x77x36x39xc
x7dx4c 3.3.2 (Hurwitz) x3x5cx6x36x2x8x2fx23 n x5x64x23x76x33
a0?n +a1?n?1 +￠￠￠+an?1?+an = 0; (3:3:4)
x58x49 a0 > 0;x2nxax4bx21x64x14x60x27Hurwitzx4bx21x64x3e
n =
flfl
flfl
flfl
flfl
flfl
fl
a1 a0 0 ￠￠￠ 0 0
a3 a2 a1 0 0
...,..,..,..,..
a2n?3 a2n?4 a2n?5 ￠￠￠ an?1 an?2
0 0 0 ￠￠￠ 0 an
flfl
flfl
flfl
flfl
flfl
fl
64
x58x49x3x6x45 i > n x69 i < 0 x19x36 ai = 0; x31x63x76x33 (3.3.4) x2x24x79xfx17x79x5ax45
x4dx6dx30x43x6d?n x2x71xax6ex7dx64x23x5cx2.
x1cx1d3.3.1 x1ax5x79x6x45x2fx23x3fxdx64?2 + a1? + a2 x2xfx17x3cx79x5ax45x4dx2
x1fx6fx13x6bx9x2fx23x17x23x5cx2,x65x5x79x6x45x2fx23x3fxdx64?3 +a1?3 +a2?+a3 x2xf
x17x3cx79x5ax45x4dx2x1fx6fx13x6bx9x2fx23x17x23x5cx2,x30a1a2 > a3.
x76 3.3.1 x61x9x39x3ax3bx76x33
x0 =?2x+y?z +x2ex;
y0 = x?y +x3y +z2;
z0 = x+y?z?ex(y2 +z2)
x5cx22x4x2x5ax36x3b.
x2a x2bx76x33x2x3ax3bx55x39x76x33x2x44x31x76x33x27
det(AI) =
flfl
flfl
flfl
2 1?1
1?1 0
1 1?1
flfl
flfl
flfl =?3 +4?2 +5?+3 = 0;
x5cx9x11x36x39 11 x2x70x10x73x7bx24x79x44x31xfx17x79x5ax45x4d,x1bx11x36x39 10 x6cx5ax6x36x76
x33x2x5cx22x4x9x20x55x5ax36x2.
2,x1ax10x78x36x391x49x47x4dx2x63x5cx44x31x76x33 det(?I?A) = 0 x2ax79x31x45x4dx3d
x79x2x45x4dxfx2x75x36,x36x3910x2ax79x53xc;x28x9x6x69x54x41x75x76,x28x29x1bx43x5cx22x4xb
x1x4fx50x5ex20,x5cx22x4x2x5ax36x3bx65x4fx50x5ex20.
x763.3.2 x71x76x33
x0 =?y; y0 = x
x2x5cx22x4x9x5ax36x2,x5cx22x4(0;0)x9x49x1b;x3dx4ex52x79x2cx57x3ax3bx4dx75x2x76x33
x0 =?y §x(x2 +y2); y0 = x§y(x2 +y2)
x2x5cx22x4(0;0)x2xbx1x24x20x27x44x4,x55x30x58x5ax36x3bx4fx17x11x76x33x2x25x5xdx2x5ex21
x55x5ex20.
x28x9x27x27x1ax8x32x33x20x7dx77,x76x33x21x27
r0 = §r3; 0 = 1:
x4fx6ct = t0x29,r(t0) = r0,(t0) = 0x2x9x9,r = r0=(1currency1(t?t0)r20)1=2; =
t?t0 + 0 x5cx9x4fx50x3cx6dx76x33x49x16+x2dx29x3cx5cx22x4x9x1bx5ax36x2x44x4x21x16?x2d
x29x3cx5cx22x4x9x20x55x5ax36x2x44x4.
x3x5cx6x64x2x28x69xbx1x2x76x33x2x5cx22x4x2x5ax36x3bx6fx7bx77x6x3x2Liapunovx19
x1ax76x7x3bx34x10.
x1ax42x6cx56x6dx57x76(Liapunov)x16x17x13x73
x10x3x7cx7dx56x34x10x36x8x2fx5(3.1.1),
x0 = f(x) (3:1:1)
65
x5cx22x4xe x2x5ax36x3b,x7cx7dx51x52f(xe) = 0,x30x1ax3ax7ex5cx22x4xe x2x28x5dx30Dx66
x6ex6fx4fx74,x57x55(3.1.1)x1ax5dx30D x66x2x9x11x7x2bx24x2cx6x4ex36.
x17x77x17x51x52x3ax1ax3ax7ex5cx22x4 xe = 0 x2x5ax30 U ‰ D x49x2x45x2bx6ex6fx22x23
V(x) x1ax4ax1bx5ax30 U?fxegx66x4fx74,x30x37x38 V(xe) = 0,x60x30x27xe x2 V x22
x23.
x62(3.1.1)x2x39x36x8x9x = x(t) x64x65 V(x) xdx1bx3 t x59x73x6c
dV
dt =
nX
k=1
@V
@xk
dxk
dt =
nX
k=1
@V
@xkfk = rV ￠f
x66x64x49x2
r = ( @@x
1; @@x
2;:::; @@x
n
)
x9x2ex30x50x7d,x72xb V(x) x2x2ex30x77x26.
x36x7
˙V =
‰ rV ￠f; x 2 U?fx
eg;
0;x = xe
x60x58x27V x22x23x27x32(3.1.1)x2x18x10x23;x6dx60x4cx73x23.
x4bx4c3.3.1 x71x13x74x75x76x33(3.1.1)x2x39x36x8x9x3xdx2x31x71x5ex(t),t? 0x2d
x5cx79x41x7fx30F x49,x40x79V x22x23x1aF x49x79x77(x69x66)x41,x5fx30
˙V? (x69?)0; 8x 2 Bg
x8x(t)x2! x8x6fx18?x39x35,x30
‰ M,= fy 2 Fj ˙V = 0g:
x6fx70 x27x31x71x5ex(t)x79x41,x58! x8x6fx18?x39x35;x6d ˙V? 0,x30V(x)x79x77
x41x29,V(x(t))x9tx2x44x21x55x79x77x41x2x22x23,x24x17
limt!+1V(x(t))(x3ax1a,x52x30) =,c:
x5cx9x3x5c? x49x14x6x5ex4ey(t),x79V(y(t)) · c,x24x17x1fx4dx5ex4ey(t),˙V · 0,x73
‰ M,(x57x39x4fxd ˙V? 0,V(x)x79x66x41x2x75x76).
x13x7dx11x46x24x74x13x7dx11x7dx4c 3.3.3 x3x5cx74x75x76x33 (3.1.1) x2x5cx22x4xe,x71
x13x3ax1axe x2V x22x23x37x38x13x6b,V(x) > 0; x 2 U?fxeg,(x28x29x60V x9x36x31
x2) ˙V? 0,(x28x29x60 ˙V x27x8x5ax2)x8x76x33(3.1.1)x2x5cx22x4xe x9x5ax36x2;x28x29x3c
x71x13x78x79
(a)x40x18x0M,= fxj ˙V = 0gx49x1bx7ex6bx5cx22x4x0 x52x6x2x15x13x5ex4e.x8x5cx22
x4x9x20x55x5ax36x2
(b)x40xe x2x14x6x5ax30N x4eM x2x26x18x49x17x7ex6bx5cx22x4x0 x52x6x2x15x13x5ex4e
x29.x5cx22x4x9x5ax36x2x3dx1bx9x20x55x5ax36x2.
(c)x6d ˙V · 0x29,x5cx22x4x9x5ax36x2x3dx1bx9x20x55x5ax36x2.
66
x6fx70 x3x5cx14x74x70x2 " > 0,x6a U x49x16x17x5cx22x4x0 x27x1bx3cx71x4cx27 " x2x7f
x4ex59x27 B"x3cx58x4ex6cx59x27 S"x3cx11V(x) x2x6ex6fx3bx1fx36x31x3b,x1a S" x66 V x4fx17x16
x4dx31x2xex70x2b:
m = min
jxj="
V(x) > 0:
x2dx11 V(0) = 0 x1f V(x) x2x6ex6fx3bx73x4fx70x5ax3ax1ax1fx75x70x2 – < ",x51x6cx6d
x 2 B–x29,x79 V(x) < m=2,x57x7fx4e B– x1ex5ax1dx2x14x6x39x36x8x9 x(t) x66
V(x(t))x9x29x13tx2x1bx4x22x23(? m=2),x6dt? 0x29,x28x9x3xdx2x5ex3ax1bx4fx50x4e
x4ex6cS" x2cx26x14x27x1ax4ex6cx66V(x)? m),x57x55x1bx4fx50x17x20B",x73xe x9x5ax36x2.
xfx6dx36x393.1.2x3cx1a B" x49x3ax1a x(t) x39x35x2 ! x8x6fx18?,x30? x9x11x40x4bx15x13
x5ex4exex72x2,x1ax28x72x5ex4ex66 V x22x23x16x8x23x2b,V(x) = c? 0,xfx6dx3fx39 3.3.1,
‰ M,x27x63x3cx6dx13x6b(a)x72x4x29,? = fxeg,x73x5cx22x4xe x9x20x55x5ax36x2.
x6dx13x6b(b)x72x4x29,x1ax5cx22x4x2x14x74x5ax30x1ex17x79x5cx22x4xe x52x6x2x5ex4ex3cx1a
x28x72x5ex4ex66 V x22x23x16x8x23x2b V(x)c > 0,x27x63x28x72x5ex4ex6d t ! +1x29,x1bx17
x42x5cx22x4xe,x73x5cx22x4xe x9x5ax36x3dx1bx9x20x55x5ax36x2.
xexdx3cx13x6bcx72x4x29x3cx1ax5cx22x4x2x14x6x1fx75x70x5ax30x1ex5ax1dx2x39x36x8x9x(t)
x66,V(x(t)) · c > 0,x24x17x6dt ! +1x29x3c x(t)x1bx4fx50x17x42x5cx22x4xe,xdx45.
x32x13x7dx11x7dx4c 3.3.4 x3x5cx74x75x76x33 (3.1.1) x2x5cx22x4 xe,x71x13x3ax1a xe
x2 V x22x23x37x38x13x6b,x71x13 V x22x23x1ax4ax1bx5ax30 U? fx0g x66x16x31x2bx2x18x0
x49x2x28x6ex27x75x1 E x2x7fx3a ˉE x3ax7ex5cx22x4 xe,x1a E x66 ˙V? 0,x5fx30x18
M,= fx 2 Ej ˙V = 0gx49x1bx7ex6bx5cx22x4xe x52x6x2x15x13x5ex4e,x8x76x33(3.1.1)x2
x5cx22x4xe x9x1bx5ax36x2.
x6f x11x5cx22x4x1bx5ax36x2x36x7x3cx24x9x6fxdx7fx3x5cx6x36x2" > 0,x4bx10 – > 0 (
– < " )x3fx63x70,x19x3ax1ax0 2 B–,x51x6c(3.1.1)x2x17x0 x27x7x2bx2x9x(t)x58x67x1a
x28x29x2at1 > 0x39x5ax7fx4eB" x52x6.
x11x36x39x2x51x52x73x7bx3cx19x3ax1ax0 2 E \B–,x51x6cV(x0) = V0 > 0,x11V x2
x6ex6fx3bxc V(0) = 0,x7bx78x3ax1a · > 0,· < –,x6dx 2?B· x29,(x58x49?B· x9x17x5c
x22x4x27x49x1bx3cx71x4cx27· x2x20x4e).x79jV(x)j < V0=2.
x7bx32xdx7,x40x3x6x45 t > 0 x79x(t) 2 B",x73x74x4dx4cx73x23 ˙V x1aE x66x9x8x31
x2,x24x17V(x(t))? V0 > 0x1bx21,x30x(t) 2 F,= EB·,x27x63,V(x(t))x1bx21
x30x79x66x41,x11x3fx39 3.3.1,x18x0fy 2 Fj ˙V = 0g ‰ M x49x7ex79 ! x8x6fx18? x49x2
x15x13x5ex4ey(t),x30y(t) x1ax4e?B· x6,x73y(t) x1bx9x5cx22x4,x28x4ex36x39x13x6bx0x1,
x57x55(3.1.1)x2x5cx22x4x9x1bx5ax36x2,x36x39xdx45.
x76 3.3.3 x6ex6fx1ax66x76x33
x0 =?y +ax3; y0 = x+ay3;
x28x29x6x36x2fx5x2x3ax3bx55x39x76x33x2x44x31xfx27? = §i,x58x5cx54x41x75x36.
x40x16x36x31x22x23 V(x;y) = 12(x2 + y2),x8x79 dVdt = a(x4 + y4),x5cx9x11x36x39
3.3.3-3.3.4,xfx6d a x2x1bx57x75x36,x4fx6cx71x77xex10xc
(i)x40 a < 0,x8 (x;y) 6= (0;0)x29,V > 0,˙V < 0,x28x29M x49x2cx6x2x15x13
x5ex4ex24x9(x;y) = (0;0)),x76x33x2x5cx22x4x9x20x55x5ax36x2;
(ii)x40 a > 0,x8 (x;y) 6= (0;0)x29,V > 0,˙V > 0,x28x29M x49x2cx6x2x15x13
x5ex4ex24x9(x;y) = (0;0),x76x33x2x5cx22x4(0;0)x9x1bx5ax36x2;
67
(iii)x40 a = 0,x8 ˙V · 0,x76x33x2x5cx22x4x9x5ax36x3dx1bx9x20x55x5ax36x2.
x763.3.4 x6ex6fx1ax66x76x33
x0 = ax?y2;y0 = 2x3y;(ax27x46x23)
x28x29x6x36x2fx5x2x3ax3bx55x39x76x33x2x44x31xfx27
= a;? = 0:
x24x17x6da > 0x29,x5cx22x4x9x1bx5ax36x2.
x3x5ca? 0x2x75x76,x16x36x31x22x23V(x;y) = x4 +y2,x8x79
dV
dt = 4ax
4;
x5cx9x11x36x393.3.3,xfx6d a x2x1bx57x75x36,x4fx6cx71x77xex10xc
(i)x40 a < 0,x8 ˙V = 4ax4? 0 x28x29M,= f(x;y)jx = 0gx49x3ax7ex2x2cx6
x2x15x13x5ex4ex24x9(x;y) = (0;0),x24x17x76x33x2x5cx22x4x9x20x55x5ax36x2;
(ii)x40 a = 0,x8 ˙V · 0,x76x33x2x5cx22x4x9x5ax36x3dx1bx9x20x55x5ax36x2.
x76 3.3.5 x6ex6fx1ax66x76x33
x0 = ax?y2;y0 = 2x3y;(ax27x46x23)
x28x29x6x36x2fx5x2x3ax3bx55x39x76x33x2x44x31xfx27
= a;
x24x17x6da > 0x29,x5cx22x4x9x1bx5ax36x2.
x3x5ca? 0x2x75x76,x16x31x36x22x23 V(x;y) = x4 +y2,x8x79
dV
dt = 4ax
4;
x5cx9x11x36x3912,xfx6d a x2x1bx57x75x36,x4fx6cx71x77xex10xc
(i)x40 a < 0,x8 ˙V = 4ax4? 0 x28x29M,= f(x;y)jx = 0gx49x2cx6x2x15x13
x5ex4ex24x9(x;y) = (0;0),x24x17x76x33x2x5cx22x4x27x20x55x5ax36;
(iI)x40 a = 0,x8 ˙V · 0,x76x33x2x5cx22x4x27x5ax36,x3dx1bx9x20x55x5ax36x2.
68
x4ax4b
x10x1fx9x38x27x38x33x3x6x2x23x41x38x39x2x41x35x0x71x2x3x41x46x6ex1f,x3x6x72x2x10x6f
x70x71x2x1ex6cx1fx41x35x1ax77x6cx6dx3ax2x31x76x2x1ax77x33x1dx2x28x9,x17x6fx5cx57x41x25x41.x63
x5cx7x48x8x4ax6x78x79x6x10x7x48x18,x3x23x4dx75x2x7x48x17x79x33x1dx2x9x1e,x7x48x18x49x33
x79*x2dx2x9x77x35x48.x33x79**x2dx2x1bx2x6fx59.
69