华东师范大学：常微分方程（精品课程）题解：odex

x0x1x2x3x4x5x6x7
x0x1x2x3x4x5x6x5x7
2003x89x9
x0x1x2x3x4x5x6x7x8x9
xaxbxc xdxexfx10x11x12x13x14x15
1.1 x0x1x2x3x4x5x6x7x8x9xa 1
1.2 xbxcxdxe 3
1.3 x7x8xfx10x11x12 14
xax16xc x17x18x19x1ax1bx1cx1d
2.1 x13x10 19
2.2 x14x15xfx10 22
2.3 x16x17x18x19x1ax0x1x2x3x1b 27
2.4 x1cx1dx19x1ax0x1x2x3 37
xax1exc x1fx18x11x20x1fx18x21x22
3.1 x7x8x9xa 44
3.2 x1ex1dx17x1fx20x21x1ax1x22 47
3.3 x14x15x23x19x1ax17x1fx24xdx20x25x21x1a 50
1
x26 x27 1.1
1.x28x29x2ax2bx16x0x1x2x3x20x1dx18,x2cx2dx2ex2fx30x31x19x1a:
1) dydx = 4x2?y:
x32,x14x1dx19x1ax2x3.
2) d
2y
dx2 +2y
dy
dx +3xy = 0:x32
,x1ex1dx23x19x1ax2x3.
3) d
2y
dx2 +p(x)
dy
dx +q(x)y = f (x)x32
,x1ex1dx19x1ax2x3.
4) dydx +cosy +x = 0.
x32,x14x1dx23x19x1ax2x3.
2,x33x34x2fx16x0x1x2x3x20xdx35x36x6x37x18x2x3x20xdx38x33x34x39x3ax35x3b
x3cx0x1x2x3x20x3dxdx3ex3fxdx35x3bx3cxbx40x11x12x35
x32,x41x42x39x43I x44x21x45x2cx46x41x39x43I x44x47x48x49x42x16x0x1x2x3x20
x4ax18x4bx4cx4dx16x0x1x2x3x41x39x43I x44x20x14x4exd,x37x18x2x3x20xdx2fx48
x49x37x18x2x3x20x4ax18x4fx18,x16x0x1x2x3x20xdx6x37x18x2x3x20xdx20x50x51
x39x3ax2f,x16x0x1x2x3x20xdx2fx41x39x43x44x21x45x20x52x0x4ax18,x36x52x53x54x38
x55x56x16x18,x57x37x18x2x3x58x59x54x5ax5bx5cx4ax18x20x5dx5ex5fx60,x14x4enx1dx16
x0x1x2x3x20x54x38nx4ex61x62x20x55x56x16x18x20xdx4bx3dxd,x3dxdx59x14x21x63
x54x2x3x64x38x20xd,x59x54x38x55x56x16x18x20xdx4bx3fxd,x5dx14x4enx1dx16x0x1
x2x3x20xd,x51x65x66x4exdx67x36x20x68x69n?1x1dx5ex18x41x42x14x6ax6bx6cx21x20
x14x6dx40,x66x6ex20x11x12x4bxbx40x11x12.
3,x6fx70x4ax18 y = 2 + cp1?x2 (x71x58 c x31x55x56x16x18)x2fx0x1x2x3?
1?x2￠ dydx +xy = 2xx20x3dxd,x2cx5dx29x48x49xbx72x73x74y(0) = 3x20xd.
xd,x75x4ax18x2x3xdx29c,x76(y? 22)=(1? x2) = c2,x77x78x79x7axx5d
x5e,x762[(1?x2)(y?2)dy=dx + (y?2)2x]=(1?x2)2 = 0;x7bx7cxfx76x0x1
x2x3 (1? x2)dy=dx + xy = 2x,(x7d,x14x15x20x2xex2fx7ex4ax18x58x20x55x56
x16x18 c xdx29,x5a x x5dx5ex7fx20x0x1x2x3x0x59x54 c x1x2x3x4xbx72x73x74:
3 = y(0) = 2+cx76c = 1,x48x49xbx72x73x74x20xdx2fy = 2+p1?x2.
4,x6fx70ey? ex = c (x66x5cx31x55x56x16x18)x2fx30x31x2x3 dydx = ex?y
x20x3dxd.
xd,x2f,x53 exp(￠) x33x34x28x18x4ax18,x6x4x2x3 exp(y)? exp(x) = c
x7x21x1x14x4ex4ax18 y(x),x8 exp(y(x))? exp(x) · c,x77x78x5a x x5dx5ex76,
exp(y(x))dy=dx?exp(x) = 0,x7cxfx7fx0x76dy=dx = exp(x?y),x8x54x38
x14x4ex55x56x16x18cx20x9x4ax18exp(y)?exp(x) = cx48x49x14x1dx0x1x2x3,xa
x21x45exp(y)?exp(x) = cx2fx3dxd.
5,x7fx5cxbx4x72x19x44x55x14x6ax20x32x19x41x77x6x7x8x52x43x20x9x1x24xc
x7ax21x77l,x2x5dx29x5fxbx4x72x19x35x48x49x20x0x1x2x3.
xd,x6(X;Y)x31x32x19x44x20x6ax6bx71x32x6a(x;y)x20x32x19x2x3x31
Y?y = y0(X?x),x36x6xx6yx8x20x5cx6ax1x3ax31(x?y=y0;0)x6
1
(0;y?xy0),x64x53x64x5dx20x2x3x31(x?y=y0)2 +(y?xy0)2 = l2.
6,x7fx5cxbx4x72x19x44x55x14x6ax20x32x19x6x4dx6ax27x3bx6ax20x0x19x52x43x20
x1dxcx25x31x16x18fi,x2x5dx29x5fxbx4x72x19x35x48x49x20x2x3.
xd,x4x12x56,tan(arctany0?arctan(y=x)) = tanfi · k,x56x4xdxcx2ex2f
x76x64x5dx2x3x31(y0?y=x) = k(1+yy0=x).
7,x5dx29x72x19x1e(x?c1)2 +(y?c2)2 = c23 x64x48x49x20x0x1x2x3,x71x58
c1,c2,c3 x31x55x56x16x18.
xd,x2x3x77x78x5axx5dx5ex14x31x762(x?c1) + 2(y?c2)y0 = 0,x3x5ax
x5dx5ex14x31x762 + 2y02 + 2(y?c2)y00,xdx29c2,c2 = y + (1 + y02)=y00,x5ax71
x79x7axx5dx5ex14x31x76x64x5dx20x0x1x2x3y0 +[(1+y02)=y00]0 = 0.
8,x14x4exbxcxdxex20xfx10x11100x12,x13x54xe10x14x15,x16x53x17x1x183x12
x20x19x1ax7dx1bx13xfx65xexfx1cx1d,x1ex1fx53x17x1x182x12x20x19x1ax20xexfx19x29.
x6xbxcx58xexfx20x21x22x41x55x3bx1fx23x24x2fx25x26x20,x5dx29x55x56x1fx23 t xb
xcx58x13xex1ax64x48x49x20x0x1x2x3x27x21xdx73x74.
xd,x6x41 t x1x18x1fx13xex1ax31 x(t) x14x15,x21xdx73x74x31xbx72x73x74:
x(0) = 10(x14x15),x41x1fx23t (x1)x1f,xfx10x11x28x29x31(100+t) (x12),xex21
x22x31 x100+t (x14x15/x12),xax12x56,x13xex1ax2ax2bx2c dxdt =?2x100+tx6bx66x0
x2fx64x5dx20x0x1x2x3.
9*,x77x6x1ex1fx41xfx58x5fx61x1fx50x51x20x69x77x4ex6bx20x42x39x6bx8x4x7ax5f
x61x21x22x23x64x28x27x20x24x13x6bT x27xfx20x25x6bD,x5ex1ex1fx20x68x22x31u,x70
x76x5fx61x21x27x1ex1fx14x26x20x27x7bx1ax31m,x5fx61x21x31x1ex1fx28x14x20x59x2ax38
x29x2ax2cx31p,x25x6bDx6u2x48x36x15x6bx2x1x62x1ex1fx68x22x20x5fx61x2x3,x28
x34,Tu = p.
xd,x6 D = ku2,k x2fx15x7ax17x18x6bx4x2bx2cx19x1ex21x2dx76x5fx61x2x3x6b
mdu=dt = p=u?ku2.
2
x26 x27 1.2
x16x18x2ax2dx2ex2f,x14x1dx19x1ax23x30x31x2x3 dx=dt + p(t)x = q(t) x20x14
x32xdx52x53x33x34x31x(t) = h(t)
c+Rtt0 q(s)=h(s)ds
·x71x58
h(t)x2fx5ax35x20
x19x1ax30x31x2x3 dx=dt + p(t)x = 0 x20x55x14x4ex36x21x20x23x24x3fxd,x52x6b
h(t) = exp(R?p(t)dt)x58x14x4ex3fx21x20x4ax18,x71x58exp(s) = es x33x34x28x18
x4ax18,cx31x55x56x16x18,x7dx56x2ex2fx58x20x77x4ex4ax18h(t)x37x38x6bx1ex14x4ex4a
x18.
1,x39x1x3ax2ax1axex5dxdx2ax2bx2x3x4fxbx40x11x12:
1) dydx +ye2x = 0
xd,y = cexp(R?e2x dx) = cexp(?e2x=2)
2) sec2 xtanydx+sec2 ytanxdy = 0
xd,x3bx2x3x52x2bx31tanydtanx+tanxdtany = 0,x75x57
d(tanxtany) = 0,x29x1x76x3dxdtanxtany = c.
3) (x+1) dydx +1 = 2e?y
xd,x7ex3bx2x3x2bx31(x+1)ey dy +(ey?2)dx = 0,x3cx57x2bx31
(x+1)d(ey?2)+ (ey?2)d(x+1) = 0,x8d[(x+1)(ey?2)] = 0,x29x1x76
x3dxd(x+1)(ey?2) = c.
4) dydx + 1yey2 +3x = 0
xd,x7ex3bx2x3x2bx316e3x dx+6ye?y2 dy = 0,x29x1x76x3dxd
2e3x?3e?y2 = c.
5) dydx = ex?y
xd,x7ex2x3x2bx31ey dy?ex dx = 0,x29x1x76x3dxdey?ex = c.
6) x2(1?y)dy +y2(1+x)dx = 0
xd,x3dxy 6= 0x1f,x7ex2x3x2bx31(1=x2 +1=x)dx+(1=y2?1=y)dy = 0.
x29x1x76x3dxd1=x+1=y +ln[y=(cx)] = 0,x3ex38x77x4ex3fxd,x = 0x67y = 0,
x36x3fx59x63x40x41x3dxdx58.
7) 3ex tanydx+(1?ex)sec2 ydy = 0; y(1) = …=4
xd,x7ex3bx2x3x2bx31?3tanyd(ex?1)+(ex?1)dtany = 0,x2x3x77x78x41
x53(ex?1)?4,x76d
h
(ex?1)?3 tany
i
= 0,x29x1x76x3dxd(ex?1)?3 tany = c,
x8tany = c(ex?1)3,xbx40x11x12x20xdx31y = arctan[[ex?1)3 =(e?1)3].
8) xp1+y2 +yp1+x2dydx = 0; y(0) = 1
x32,x3dxdx31p1+x2 +p1+y2 = c.
xbx40x11x12x20xdx31y =
r?p
1+x2?1?p2
·2
1.
9) (1+x)ydx+x(1?y) dy = 0; y(2) = 0
x32,xbx40x11x12x20xdx31y = 0 (x59x42x75x3dxdln((xy)=c) = y?xx58x76
x69).
10) xy
1+x2
· dy
dx = 1+y
2; y(1) = 0.
xd,x7ex2x3x2bx31d(1+y2)=d(x2) = (1+y2)=[x2(1+x2)],x1x3ax2ax1ax76
d(1+y2)=(1+y2) = [1=x2?1=(1+x2)]d(x2),x29x1x76x3dxdx31
3
(1 + x2)(1 + y2) = cx2,xbx40x11x12x20xdx31(1 + x2)(1 + y2) = 2x2 xdx29y
x76y = §[(x2?1)=(x2 +1)]1=2.
2,x7ex2ax2bx2x3x2bx31x2ax1ax1x3ax2x3x7fx5dxd:
1) (x+y) dx?(x?y) dy = 0
xd,x7ex2x3x77x78x41x532,x3x43x44x1bx45x2bx31
2(xdx+ydy)?2(xdy?ydx) = 0,x52x46,x52x47x48x0x1
d?x2 +y2￠?2(x2 +y2)darctan(y=x) = 0,x77x78x1ex49x53x2 +y2 x76:
d(x2 +y2)=(x2 +y2)?2darctan(y=x) = 0,x29x1x76x3dxd:
ln(x2 + y2)?2arctan(y=x) = c,(x7d,x8x12x2fx30x31x2x3,x4ax52xax30x31x2
x3x20x3dx16xdxex5dxd,x4bx4cx4d).
2) y2 dx+(x2?xy)dy = 0
xd,x7ex2x3x43x44x1bx45x2bx31?y(xdy? ydx) + x2 dy = 0,x47x0x1x76
x2yd(y=x)+x2 dy = 0,x3dxy 6= 0x1f,x77x78x1ex49x53x2yx76:
d(y=x)+dy=y = 0,x29x1x76x3dxd,?y=x+ln(y=c) = 0,x4fx2bx31
y = cexp(y=x);x3ex38x3fxdx = 0x59x63x40x41x3dxdx58.x3fxdy = 0x52x53x63
x54x41x3dxdx20x7fx14x4ex4fx2fx58.
(x7d,x8x12x2fx30x31x2x3,x4ax52xax30x31x2x3x20x3dx16xdxex5dxd).
3) dydx = 2y
2?xy
x2?xy +y2
xd,x2x3x2fx30x31x2x3,x13x3cx44x20x5bx5cx4ax18u,x48x49x79x17x2fy = xu,
x5a x x5dx5ex76x79x17x2f dy=dx = u + xdu=dx,x7ex66x77x2fx37x1bx2x3,x76:
u+xdu=dx = (2u2?u)=(1?u+u2),x1x3ax2ax1ax76:
[2=(u?1)?1=u?3=(u?2)] du = 2dx=x,x29x1x76
ln[(u?1)2=(cu(u?2)3)] = lnx2,x4fx2bx31(u?1)2 = cx2u(u?2)3,x53u = y=x
x37x1bx76x3dxd(y?x)2 = cy(y?2x)3,x3ex38x77x4ex3fxdy = 0,x67y = 2x,x36
x3fx59x63x40x41x3dxdx58,x1x3ax5ax35x7au = 0x27u = 2 (x7d,x6u = 1x5ax35x20
xdy = xx52x53x63x54x41x3dxdx58(c = 0x1f)).
4) xdy=dx = xexp(y=x)+y +x
xd,x2x3x2fx30x31x2x3,x39x2ax1ax37x50 y = xu,x76x2ax1ax1x3ax2x3
xdu=dx = exp(u)+1,x3cx57x51x48dx=x+dexp(?u)=(exp(?u)+1) = 0,x29x1
x76,ln(x(exp(?u)+1)=c) = 0,x37x52x3bx2ax1ax76x3dxdx(1+exp(?y=x)) = c,
5) x(lnx?lny)dy?ydx = 0
xd,x2x3x2fx30x31x2x3,x39x2ax1ax37x50 y = xu,x76x2ax1ax1x3ax2x3:
xdu=dx =?u(1+lnu)=lnu,u 6= 1=ex1fx6bx2bx31:
dx=x+lnudlnu=(1+lnu) = 0,x29x1x76ln[cxu=(1+lnu)] = 0,x37x52x3bx2a
x1ax76x3dxdcy = 1+lny?lnx,x3fxdy = x=ex63x54x41x3dxdx58.
6) dy=dx = (2x?y +1)=(x?2y +1)
xd,x7ex2x3x2bx31x0x1x4fx2fx2cx1x1bx76:
[(2x+1)dx+(2y?1)dy]?(xdy +ydx) = 0,x3cx57x76
d(x2 +x+y2?y)?d(xy) = 0,x29x1x76x3dxd:
x2+y2+x?y?xy = c,(x7d,x8x12x4ax52x2bx31 dydx = 2(x+1=3)?(y?1=3)(x+1=3)?2(y?1=3)
x7fxax30x31x2x3x20xdxex54x5dxd,x4bx4cx4d).
7) dy=dx = (2x+3y +4)=(4x+6y +5)
4
xd,x55u = 2x + 3y,x56du=dx = 2 + 3dy=dx = 2 + 3(u + 4)=(2u + 5),
x8du=dx = (7u+22)=(2u+5),7u+22 = 0x1f,x76x3fxd14x+21y+22 = 0,
7u+22 6= 0x1f,x1x3ax2ax1ax76[2?9=(7u+22)]du = 7dx;,x29x1x76,
2u?9=7ln[(7u+22)=c] = 7x,x37x52x3bx2ax1ax7cxfx76x3dxd
7(2y?x)?3ln[(14x+21y +22)=c] = 0,x57x14x4fx2fx31
14x + 21y + 22 = cexp(7(2y?x)=3),x3fxd14x + 21y + 22 = 0x63x54x41x3d
xdx20x7fx14x4fx2fx58.
8) dy=dx = (x+1)2 +(4y +1)2 +8xy +1
xd,x52x46 dy=dx = (x + 4y + 1)2 + 2,x56x55 u = x + 4y + 1,x75x57
du=dx = 1+4dy=dx = 1+4(u2 +2) = 4u2 +9,x8du=dx = 4u2 +9,x1x3a
x2ax1ax763d(2u)=(4u2 + 9) = 6dx,x29x1x76,arctan(2u=3) = 6x + c,x37x52x3b
x2ax1ax7cxfx76x3dxdarctan((2x+8y +2)=3) = 6x+c.
9) dy=dx = (y6?2x2)=(2xy5 +x2y2)
xd,x7ex3bx2x3x2bx31d(y3)=dx = 3[(y3)2?2x2]=(2xy3 +x2),x55v = y3,
x76x30x31x2x3 dv=dx = 3(v2? 2x2)=(2xv + x2),x56x55 v = xu,x5a x x5d
x5ex76 dv=dx = u + xdu=dx = 3(u2? 2)=(2u + 1),x8x76x2ax1ax1x3ax2x3
xdu=dx = (u?3)(u+2)=(2u+1),x1x3ax2ax1ax76
[7=(u?3)+3=(u+2)]du = 5dx=x,x29x1x767lnju?3j+3lnju+2j = 5ln(cx).
x37x52x3bx2ax1ax76x3dxd(y3?3x)7(y3 + 2x)3 = cx15,(x7d,x5ax35x7au = 3x67
u =?2x20x3fxdx63x54x41x3dxdx58).
10) dy=dx = (2x3 +3xy2 +x)=(3x2y +2y3?y).
xd,x7ex3bx2x3x2bx31
d(y2)=d(x2) = [2(x2? 1) + 3(y2 + 1)]=[3(x2? 1) + 2(y2 + 1)],x75x57x52x55
u = x2?1,v = y2 +1,x3bx2x3x2bx31x30x31x2x3dv=du = (2u+3v)(3u+2v),
x55v = uw,x5aux5dx5ex76,dv=du = w + udw=du = (3w + 2)=(2w + 3),x8
x76x2ax1ax1x3ax2x3udw=du = 2(1?w2)(2w +3),x1x3ax2ax1ax76
[1=(w +1)?5=(w?1)]dw = 4du=u,x29x1x76:
lnjw +1j?5lnjw?1j = 4lnjuj+c,x37x52x3bx2ax1ax7cxfx76x3dxd
(y2?x2 +2)5 = c(x2 +y2),(x7d,x5ax35x7aw = 1x20x3fxdx63x54x41x3dxdx58)
3,x39x16x18x2ax2dx2ex2fx5dxdx2ax2b(x52x2bx31x2x19x1ax2x3x4fBernoullix2
x3x20x3dxdx4fxbx40x11x12:
1) dy=dx = y +sinx
xd,x6bx19x1ax30x31x2x3dy=dx?y = 0x20x14x4ex3fxdh(x) = exp(x),x35
x39x16x18x2ax2dx2ex2fx76:
y = exp(x)[c+R sinxexp(?x)dx] = cexp(x)?(sinx+cosx)=2.
2) dx=dt = exp(2t)?3x
xd,x6bx5ax35x20x30x31x2x3x20x14x4ex3fxdx31h(t) = exp(?3t),x35x39x16x18
x2ax2dx2ex2fx76,x = exp(?3t)[c+R exp(5t)dt] = cexp(?3t)+exp(2t)=5.
3) dy=dx?ny=x = xn exp(x)
x32,y = xn(c+exp(x)).
4) dy=dx+(1?2x)y=x2?1 = 0
xd,x6bx5ax35x20x30x31x2x3x20x14x4ex3fxdh(x) = x2 exp(1=x),x35x39x16x18
x2ax2dx2ex2fx76y = x2 exp(1=x)[c+R exp(?1=x)d(?1=x)] = x2[cexp(1=x)+1].
5) dy=dx = ytanx+cosx
5
xd,x6bx5ax35x20x30x31x2x3x20x14x4ex3fxdx31 h(x) = 1=cosx = secx,x35
x39x16x18x2ax2dx2ex2fx76y = secx[c+R cos2 xdx] = [(x+2c)secx+sinx]=2.
6) dy=dx?y = 2xexp(2x); y(0) = 1
x32,x3dxdx31y = cexp(x)+2(x?1)exp(2x),xbx40x11x12x20xdx31
y = 3exp(x)+2(x?1)exp(2x).
7) xylnydx+?x2?lny￠ dy = 0
xd,x2x3x77x78x1ex412=yx2x3x2bx31lnyd(x2) + 2(x2?lny)dlny = 0,
y 6= 1x1f,x3cx57x2bx31x19x1ax2x3d(x2)=dlny + 2x2=lny = 2,x58x39x16x18x2a
x2dx2ex2fx76x3dxd,x2 = c=ln2 y +2=3lny,x3fxdy = 1x59x63x54x41x3dxdx58.
8)dy=dx+2y=(x+1) = (x+1)3
x32,y = c(x+1)?2 +(x+1)4=6.
9)x1ex7a1.6,x5e
10) dy=dx+xy = x3y3
xd,x2fBernoullix2x3,x3dy 6= 0x1f,x59x7ex36x2bx31x19x1ax2x3
d(y?2)=dx = 2xy?2?2x3,x35x39x16x18x2ax2dx2ex2fx76x3dxdx31
y?2 = cexp(x2)+x2 +1,x3ex38x3fxdy = 0 (x59x63x54x41x3dxdx58).
11) dy=dx = 1=(xy +x3y3)
xd,x7ex5ax2ax1ax6x5bx2ax1ax5cx50x76Bernoullix2x3 dxdy = xy + x3y3,x7e
x36x2bx31x19x1ax2x3d(x?2)=dy =?2yx?2?2y3,x75x57x35x39x16x18x2ax2dx2ex2f
x76x3dxd,x?2 = cexp(?y2)+1?y2.
12) dy=dx = x?2(3x+exp(y))
xd,x7ex2x3x2bx31x19x1ax2x3dexp(?y)=dx+3exp(?y)=x =?1=x2,x35
x39x16x18x2ax2dx2ex2fx3cx57x76x3dxdexp(?y) = cx?3?(2x)?1.
13) dy=dx = (x4 +y3)=(xy2)
xd,x2fBernoullix2x3,x52x2bx31x19x1ax2x3d(y3)=dx?3y3=x = 3x3,x29
x1x76x3dxd,y3 = cx3 +3x4.
14) dy=dx = 1=(xcosy +sin2y)
xd,x7ex5ax2ax1ax6x5bx2ax1ax5cx50x76x19x1ax2x3dx=dy = xcosy +sin(2y),
x6bx5ax35x20x30x31x2x3x20x14x4ex3fxdx31x = h(y) = exp(siny),x75x57x35x39x16
x18x2ax2dx2ex2fx76x = 2exp(siny)[c +R sinyexp(?siny)dsiny],x29x1x76x3d
xd,= cexp(siny)?2(1+siny).
4,x58x39x5dx0x1x2x3(x121–6,12)x27x39x29x1x5bx64x2xe,(x127–11)x5dx29x2a
x2bx2x3x20xd
1)?x2 +y￠ dx+(x?2y) dy = 0
xd,x7ex2x3x1x1bx31 (x2 dx? 2ydy) + (ydx + xdy) = 0,x47x0x1x76
(x2 dx?2ydy)+d(xy) = 0,x29x1x76x3dxd,x3=3?y2 +xy = c.
2) exp(?y)dx+(1?xexp(?y))dy = 0
xd,x7ex2x3x1x1bx31(exp(?y)dx?xexp(?y)dy)+ dy = 0,x47x0x1x76
d(xexp(?y))+dy = 0,x29x1x76x3dxd,xexp(?y)+y = c.
3) (y?3x2)dx?(4y?x)dy = 0
xd,x7ex2x3x1x1bx31 (ydx + xdy)? (3x2 dx + 4ydy) = 0,x47x0x1x76
d(xy)?d(x3 +2y2) = 0,x29x1x76x3dxd,xy?x3?2y2 = c.
4) (9x2 +y?1)dx?(4y?x)dy = 0
6
xd,x7ex2x3x1x1bx31[(9x2?1)dx?4ydy]+(ydx+xdy) = 0,x47x0x1
x76d(3x3?x?2y2)+d(xy) = 0,x29x1x76x3dxd,3x3?x?2y2 +xy = c.
5) [y?1 sin(x=y)?yx?2 cos(y=x)+1]dx
+[x?1 cos(y=x)?xy?2 sin(x=y)+y?2]dy = 0
xdxex14,x5eM(x;y) = y?1 sin(x=y)?yx?2 cos(y=x)+1,
N(x;y) = x?1 cos(y=x)?xy?2 sin(x=y)+y?2,
x52x76 @M(x;y)=@y = @N(x;y)=@x,x5bx5fx2x3x2fx60x3dx20,x6x71x29x1
x31U(x;y) = c,x61@U(x;y)=@y = N(x;y),x79x7ayx29x1,x76
U(x;y) =
Z
[x?1 cos(y=x)?xy?2 sin(x=y)+y?2]@y
= sin(y=x)?cos(x=y)?1=y +c(x)
x71x58c(x)x2fx62x21x20xx20x4ax18,x31x5dc(x),x58x39x47xcx2f
@U(x;y)=@x = M(x;y),x52x76c0(x) = 1,x56x52x6bc(x) = x,x64x53x29x1x31
U(x;y) = c,x71x58U(x;y) = sin(y=x)?cos(x=y)?1=y +x.
xdxex1ex4ex7ex0x1x2x3x1bx45x31[sin(x=y)=ydx?xsin(x=y)=y2 dy]
+[?ycos(y=x)=x2 dx+cos(y=x)=xdy]+[dx+1=y2 dy] = 0,x47x0x1,x29
x1x76x4e?cos(x=y)+sin(y=x)+x?1=y = c
6) 2x(yexp(x2)?1)dx+exp(x2)dy = 0
xd,x7ex3bx2x3x2bx31[yd(exp(x2))+exp(x2)dy]?2xdx = 0,x47x0x1x76
d(yexp(x2))?d(x2) = 0,x29x1x76x3dxdyexp(x2)?x2 = c,x4fxdx29x63x4ax18
x4fx2f,y = (c+x2)exp(?x2).
7) (exp(x)+3y2)dx+2xydy = 0
xd,x7ex2x3x1x1bx48(3y2 dx+2xydy)+exp(x)dx = 0,x47x0x1x76
x?2 d(x3y2)+exp(x)dx = 0,x52x46x29x1x5bx64x52x6bx31x2,x75x57x2bx48x5dx0x1
x2x3d(x3y2)+x2 exp(x)dx = 0,x29x1x76x3dxdx3y2+exp(x)(x2?2x+2) = c.
8) (x2 +y2 +x)dx+xydy = 0
xd,x7ex2x3x1x1bx48(x2 +x)dx+(y2 dx+xydy) = 0,x47x0x1x76
(x2+x)dx+(2x)?1 d(x2y2) = 0,x52x46x29x1x5bx64x52x6bx3112x,x75x57x2bx48x5d
x0x1x2x312x(x2+x)dx+6d(x2y2) = 0,x29x1x76x3dxd3x4+4x3+6x2y2 = c.
9) (x+2y) dx+xdy = 0
xd,x7ex2x3x1x1bx48xdx+(2ydx+xdy) = 0,x47x0x1x76
xdx+x?1d(x2y) = 0,x52x46x29x1x5bx64x52x6bx313x,x75x57x2bx48x5dx0x1x2x3
3x2 dx+3d(x2y) = 0,x29x1x76x3dxdx3 +3x2y = c.
10) (2xy2 +y)dx?xdy = 0
xd,x7ex2x3x1x1bx482xy2 dx+(ydx?xdy) = 0,x47x0x1x76
2xy2 dx + y2 d(x=y) = 0,x52x46x29x1x5bx64x52x6bx31y?2,x75x57x2bx48x5dx0x1
x2x32xdx+d(x=y) = 0,x29x1x76x3dxdx2 +x=y = c,x3ex38x3fxdy = 0x59
x63x54x41x3dxdx58.x36x2fx3bx2x3x77x78x49x53x24x57x65x66x20xd.
11) [y?x(x2 +y2)]dx?xdy = 0
xd,x7ex2x3x1x1bx48?x(x2 + y2)dx? (xdy? ydx) = 0,x47x0x1x76
x(x2 + y2)dx? (x2 + y2)darctan(y=x) = 0,x52x46x29x1x5bx64x52x6bx31
2=(x2 +y2),x75x57x2bx48x5dx0x1x2x32xdx+2darctan(y=x) = 0,x29x1x76
x3dxdx2 +2arctan(y=x) = c.
7
12) 2xy?3 dx+y?4(y2?3x2)dy = 0.
xd,x7ex2x3x2bx31 [y?3 d(x2) + x2 d(y?3)]? d(y?1) = 0,x47x0x1x76
d(y?3x2)?d(y?1) = 0,x29x1x76x3dxdx2y?3?y?1 = c.
5,x5dxdx2ax2bx9x2x3:
1) y02?3y0 +2 = 0
xd,x1xdx5bx2fx76(y0?1)(y0?2) = 0,x56x4y0 = 1,x76x3dxdy = x+c,
x4y0 = 2,x76x3dxdy = 2x+c.
2) y = 2xy0 +x2y04
xd,x13x3cx67x18p = y0,x2x3x52x51x48x67x18x4fx2f
y0 = p; (1)
y = 2xp+x2p4; (2)
x31x2ex45x2ax1ay,x7e(2)x2fx5axx5dx5ex7fx11x45(1)x2fx6bx76p,x6xx20x0x1
x2x32p+2xdp=dx+2xp4 +4x2p3 dp=dx?p = 0,x7cxfx76
(1 + 2xp3)(p + 2xdp=dx) = 0,x41 + 2xp3 = 0,xdx29p =?(2x)?1=3,x37
x1b(2)x2fx76x3fxdy =?3=4(4x2)1=3,x4p+2xdp=dx = 0,x29x1x76
p = c(§x)?1=2,x37x1b(2)x2fx76x3dxd,y = 2c(§x)1=2 + c4,(x7dx56:x5dx29px7f
x59x42x37x1b(1)x2fx3x29x1,x30x61x68x76x69x14x4ex59x2f“x55x56x20”x16x18x20xd,x5b
x31x76x69x20xdx3ex37x38x48x49(2)x2f)
3) xy03 = 1+y0
xd,x55dy=dx = 1=t,x61x76x67x18x4fx2fx20x0x1x2x3
x = t3 + t2,dx=dy = t,x31x2ex45x2ax1ax,x7ex6cx2fx5ayx5dx5ex7fx11x45x7fx2fx6b
x76t,x6yx20x0x1x2x3(2t2+2t)dt=dy?t = 0,x8 dy = (3t+2)dt,x29x1x76
y = 3t2=2+2t+c,x75x57x76x67x18x4fx2fx20x3dxd,x = t3+t2; y = 3t2=2+2t+c.
4) y02 +(x+y)y0 +xy = 0
xd,x1xdx5bx2fx76(y0 +x)(y0 +y) = 0,x56x4y0 =?x,x76x3dxd
y =?x2=2+c,x4y0 =?y,x76x3dxdy = cexp(?x).
5) y = 2xy0 +x2=2+y02
xd,x13x3cx67x18p = y0,x2x3x52x51x48x67x18x4fx2f
y0 = p; (1)
y = 2xp+x2=2+p2 = (p+x)2?x2=2,(2)
x31x2ex45x2ax1ay,x7e(2)x2fx5axx5dx5ex7fx11x45(1)x2fx6bx76p,x6xx20x0x1
x2x32(p + x)(dp=dx + 1)?x?p = 0,x7cxfx76(x + p)(2dp=dx + 1) = 0,
x4 p =?x,x37x1b(2)x2fx76x3fxd,y =?x2=2,x42dp=dx + 1 = 0,x29x1x76
p =?x=2+c,x37x1b(2)x2fx76x3dxd,y = (x=2+c)2?x2=2.
6) y = xy0 +y0?y02
xd,x2f Clairaut x2x3,x56x3dxdx31 y = cx + c? c2,x77x78x3x79x7a c
x5dx5ex76,0 = x + 1? 2c,x8 c = (x + 1)=2,x37x1bx3dxdx33x69x2fx76x3fxd
y = (x+1)2=4.
7) y02 +2xy0 +2y = 0
8
xd,x13x3cx67x18p = y0,x2x3x52x51x48x67x18x4fx2f
y0 = p; (1)
y =?xp? 12p2; (2)
x31x2ex45x2ax1ay,x7e(2)x2fx5axx5dx5ex7fx11x45(1)x2fx6bx76p,x6xx20x0x1x2
x3?p?xp0?pp0?p = 0,p 6= 0x1f,x7ex36x6ax51x48x53px31x5ax2ax1ax20x19x1ax2
x3dx=dp =?x=(2p)?1=2,x36x20x3dxdx2fx = c(§p)?1=2?p=3,x37x1b(2)x76
x67x18x4fx2fx20x3dxdx = c(§p)?1=2?p=3; y = currency1c(§p)1=2?p2=6,x3dp = 0
x1fx6bx76x3fxdy = 0.
8) x2 +y02 = 1
xd,x55x = sint,t 2 [?…=2;…=2],x76x67x18x4fx2fx20x0x1x2x3
x = sint; y0 = §cost,x2ex45x,dy = y0dx = §cos2 tdt,x29x1x76x67x18
x4fx2fx20x3dxd,x = sint; y = §[2t+sin(2t)]=4+c.
9) y03 +y3?3yy0 = 0
xd,x55y = y0t,x37x1bx2x3x76y0 = 0x4fy0 = 3t=(1 + t3),x75x6cx6dx76x3f
xdy = 0,x75x7fx6dx76x67x18x2x3x1by = 3t2=(1 + t3),y0 = 3t=(1 +t3),x3x4
dx = dy=y0 = (1+t3)=(3t)dy = [?1+3=(1+t3)]dt
= [?1+1=(1+t)?(t?1=2)=(t2?t+1)+3=(2(t2?t+1))]dt:
x29x1x76x67x18x4fx2fx20x3dxd:
8>
><
>>:
x =?t+ln
§(1+t)
pt2?t+1
+p3arctan
2t?1
p3
+c;
y = 3t
2
1+t3
10) y = exp(y0)y02
xd,x13x3cx67x18p = y0,x2x3x52x51x48x67x18x4fx2f
y0 = p; (1)
y = exp(p)p2,(2)
x31x2ex45x2ax1ay,x7e(2)x2fx5axx5dx5ex7fx11x45(1)x2fx6bx76p,x6xx20x0x1x2
x3x56(exp(p)p2 + 2pexp(p))p0?p = 0,x7cxfx76p(1?exp(p)(2 + p)p0) = 0,
x4p = 0,x37x1bx532x2x2fx76x3fxdy = 0,x4(1?exp(p)(2 +p)p0) = 0x76x67x18
x4fx2fx20x3dxd
x = (p+1)exp(p)+c; y = p2 exp(p):
6,x7fx5cf(x) > 0x41(0;+1)x44x0x1x46f (x)
Z x
0
f (t) dt = 1;x > 0,x2
x5df (x)x20x33x69x2f.
xd,x7ex2x3x2bx31
Z x
0
f (t) dt = 1f (x),x77x78x5a x x5dx5ex76,f (x) =
f
0(x)
f2 (x),x29x1x76f
2 (x) = 1
2x+c,x37x1bx3bx2x3x76c = 0,x56f (x) =
1p
2x.
9
7,x77x6x0(0)x3x41,x2x5dx48x49
x(t+s) = x(t)+x(s)1?x(t)x(s)
x20x4ax18x(t).
xd,x55t = 0,s = 0,x76x(0) = 0,x5bx5f
x0(t) = lims!0 x(t+s)?x(t)s
=
h
1+x2 (t)
i
lims!0 x(s)s[1?x(t)x(s)]
=
h
1+x2 (t)
i
lims!0 x(s)s
=
h
1+x2 (t)
i
x0(0)
.
x8x(t)x48x49x0x1x2x3x0(t) = x0(0)￡1+x2 (t)?,x29x1x76
arctan(x) = x0(0)t+c,x8x(t) = tan(x0(0)t+c),x3x4x(0) = 0x76c = 0,
x56x76x64x5dx4ax18x31x(t) = tan(x0(0)t).
8,x5dx14x72x19,x65x76x41x36x44x4x55x14x6ax20x32x19x5x7ax6x7x8x43x20x9x1
xax32x6ax64xbx1.
xd,x6x41x68xcx6x7x17xOyx58x72x19x20x2x3x31y = y(x),x41x6a(x;y(x))
x20x32x19x6x x8,y x8x1bx48x20xdxcx4fx58,x4x12x56xdxcx4fx20x1cxcx7a 2jyj,
xexcx7a2jxj,x5bx5fxdydx =?y,x29x1x76xy = c,c 6= 0.
9,x6x4ax18x(t)x41(?1;+1)x44x0x1,x(t)x59x47x31x24,x0(0)x3x41,
x46x48x49x73x74x(t+s) = x(t)x(s);x2x5dx5fx4ax18.
xd,x55t = 0,s = 0,x76x(0) = 0,x4fx(0) = 1,x52x46x3dx(0) = 0x1f,
x(t) · x(t)x(0) · 0,x56xfx10x11x(0) = 1x20x12x13,
x0(t) = lims!0 x(t+s)?x(t)s = x(t) lims!0 x(s)?1s = x0(0)x(t),
x8x(t)x48x49x0x1x2x3x0(t) = x0(0)x(t),x29x1x76x(t) = cex0(0)t,
x3x4x(0) = 1x76c = 1,x56x64x5dx4ax18x31x(t) = ex0(0)t.
10.x51x29x2x3M (x;y) dx+N (x;y) dy = 0x14x38x4fx31?(x§y),?(xy),
x2 §y2￠x20x29x1x5bx64x20x15x51x73x74.
x32,x5bx31M (x;y) dx+N (x;y) dy = 0x14x38x4fx70?(’(x;y))x20x29x1
x5bx64x20x15x51x73x74x2f:
My?Nx
N`x?M`y
x16x2f’x20x4ax18,x64x53x2x3x38x4fx70?(x§y),?(xy),x2 §y2￠x20x29x1
x5bx64x20x15x51x73x74x2f
My?Nx
N currency1M,
My?Nx
Ny?Mx,
My?Nx
Nxcurrency1My,x1x3ax16x2fx § y,xy,x
2 § y2 x20x4a
x18.
11,x6M (x;y);N (x;y)x24x2fx,yx20x0x1x52x0x20mx31x30x31x4ax18,
m 6=?1,x5eU (x;y) = xM (x;y)+yN (x;y),x70x17
10
1) xMx +yMy · mM (x;y); xNx +yNy · mN,
2)x18M (x;y) dx+N (x;y) dy = 0x31x5dx0x1x2x3,x61x71x3dxdx31
U (x;y) = c.
3)?(x;y) = 1U (x;y) x2fx2x3M (x;y) dx + N (x;y) dy = 0x20x29x1x5b
x64.
x70,1)x5bM (x;y)x2fx,yx20mx31x30x31x4ax18,x8x5ax7ax55x3bt > 0,x48
x62x47xcx2fM (tx;ty) · tmM (x;y),x4M x20x52x0x1a,x77x78x5atx5dx5e,x76
xM01 (tx;ty)+yM02 (tx;ty) · mtm?1M (x;y),x71x58M01 (tx;ty)x3e M02 (tx;ty)
x1x3ax33x34x4ax18x5ax19x14x3ex19x1ex4ex5ax2ax1ax5dx1ax5ex18.x55x44x2fx58t = 1x8
x76x70,x1exfx52x70x79x7aN (x;y)x20x47xcx2f.
2)x5bm 6=?1,x3bx0x1x2x3xcx1bx7a(1+m)(M dx+N dy) = 0,
x1cx5bM dx+N dy = 0x31x5dx0x1x2x3,x64x53My = Nx,
x61dU = M dx+N dy +(xMx +yNx) dx+(xMy +yNy) dy,
x1cx5bM dx+N dy = 0x31x5dx0x1x2x3,x38My = Nx,x64x53
dU = M (x;y) dx+M (x;y) dy +(xMx +yMy) dx+(xNx +yNy) dy,
x3x5bM (x;y),N (x;y)x24x2fx,yx20mx31x30x31x4ax18,x64x53x41)x76
dU = (1+m)(M dx+N dy),x8U (x;y) = cx2fx2x3x20x3dxd.
3)x4x29x1x5bx64x20x21x45,xfx51x70x17 @(?M)@y? @(?N)@x = 0x8x52.
x5dx5ex76
@(?M)
@y?
@(?N)
@x =
1
U2 [U (My?Nx)+NUx?MUy]; (2)
x71x58
Ux = M +xMx +yNx; Uy = N +xMy +yNy; (3)
x7e(3)x37x1b(2)x7cxfx76
@(?M)
@y?
@(?N)
@x =
1
U2 [N (xMx +yMy)?M (xNx +yNy)]; (4)
x3x7e(1)x37x1b(4)x76 @(?M)@y? @(?N)@x = 0,x70x1d.
12,x7fx5cx2ax2bRiccatix2x3x20x14x4ex3fxd’(x),x2x5dx29x71x3dxd:
1) y0e?x +y2?2yex = 1?e2x; ’(x) = ex,
xd,x55y = ex +u?1,x37x1bx2x3x76x5bx5cx4ax18ux20x2x3u0 = ex,x29x1
x76u = ex +c,x5bx5fx3bx2x3x20x3dxdx31y = ex +(ex +c)?1.
2) y0 +y2?2ysinx = cosx?sin2 x; ’(x) = sinx,
xd,x55y = sinx + u?1,x37x1bx2x3x76x5bx5cx4ax18ux20x2x3u0 = 1,x29
x1x76u = x+c,x5bx5fx3bx2x3x20x3dxdx31y = sinx+(x+c)?1.
3) 4x2(y0?y2) = 1; ’(x) =? 12x,
xd,x55y =?(2x)?1 +u?1,x37x1bx2x3x76x5bx5cx4ax18ux20x2x3
u0 = x?1u?1,x29x1x76u =?xln(cx),x5bx5fx3bx2x3x20x3dxdx31
y =?(2x)?1?(xln(cx))?1.
11
4) x2y0 +(xy?2)2 = 0; ’(x) = 1x,
xd:x55y = x?1+u?1,x37x1bx2x3x76x5bx5cx4ax18ux20x2x3u0 =?x?1u+1,
x29x1x76u = x?1(c+x2=2),x5bx5fx3bx2x3x20x3dxdx31y = x?1+x(c+x2=2)?1.
5) y0 = (x?1)y2 +(1?2x)y +x; ’(x) = 1:
xd,x55y = 1+u?1,x37x1bx2x3x76x5bx5cx4ax18ux20x2x3u0 = u+1?x,
x29x1x76u = ex(c+xe?x),x5bx5fx3bx2x3x20x3dxdx31y = 1+(cex +x)?1.
13,x5dx29logisticx2x3x20xbx40x11x12
dx
dt = rx(1?
x
xf ); x(0) = x0 6= 0; (r > 0;xfx2fx16x18)
xdx(t),x2cx60x29 limt!+1x(t)x20x40x27x1ex17x71x1fx20x56x45.
xd,x66x2fn = 2x1fx20Bernoullix2x3,x49x1xbx21xdx · 0x22,x71xdx52
x51x48x4fx2f
x(t) = xf
1?
1? xfx
0
e?rt
x56x52x46 limt!+1x(t) = xf,x1ex17x1xfx51 x(0) 6= 0,x55x3bxd x x23x24xbx25x26
x = xf.
14*,x5dx29xbx40x11x12
dk
dt = sf(k)k; k(0) = k0
x20xdk(t;s),x71x58f(k) = akfl x31Cobb-Douglasx27x28x4ax18,x66x5a > 0,
0 < fl < 1x25x31x16x18,x60x29k1(s) = limt!1k(t;s)x27x1fx43x29x2ax2bx2cx1fx20
x2dx25x2ex2fc1(s) = (1?s)f(k1(s)),x4x5fx5dx29x65x76c1(s)x69x69x30x2cx40
x20sx40,x75x57x6fx70x31x8x29x32x20x33x34x35x61f0(k1(s)) =?x20x36x36x1a.
xd,x66x2f n = fl x1fx20 Bernoulli x2x3,x49x1x29x1fx20x56x45x20xbx21xd
k · 0x22x31
k1?fl(t;s) = k1?fl0 e(1?fl)t + as?
1?e(1?fl)t
·
:
x64x53
k1(s) = limt!1k(t;s) =
as
1
1?fl ;
c1(s) = (1?s)f(k1(s)) = a(1?s)
as
fl
1?fl,
x5dx36x20x30x2cx40x52x76,x3ds = flx1fx6bx30x2cx40maxc1(s) = a(1?fl)
afl
· fl
1?fl,
xbx2dx6fx70x31x8x29x32x20x33x34x35x61f0(k1(s)) =?x20x36x36x1a.
15* x41x73x121.1x20x19x2ex12x58,x3dx1ex1fx75x2fx30x31x26x32x72x5fx61x1f,x5dx29
x71x5fx61x68x22ux6x1fx43tx20x79x17x6bx2cx1ex17x71x1fx20x56x45.
12
xdx4ex5ea = (p=k)1=3,x61x2x3x52x2bx31
[? 2u?a + (2u+a)?3au2 +au+a2 ]du = 6kam dt:
x29x1x2cx58x39xbx72x73x74u(0) = 0,x76xdx31
ln u
2 +au+a2
(u?a)2 +
…p
3?2
p3arctan(2u+ap
3a ) =
6kat
m,
x52x46x6bx3dt ! +1x1fx6b u ! a,x1ex17x1x30x7fxcx23x68x22x20x2bx6fx6bx25x6b
x2bx2cx6bx24x13x6bx11x12x6bx30x50x23x7axbx25x6bx68x22x23x7ax14x4ex21x40a.
13
x26 x27 1.3
1,x5dx29xbx40x11x12y0 = x+y2;y(0) = 0x20x19xdx31x37x38xd’3 (x).
x32,’3 (x) = x
2
2 +
x5
20 +
x8
160 +
x11
4400.
2,x70x17Bellmanx59xcx2f,x6x16x18k > 0,f (x)? 0,x27’(x)x41[fi;fl]
x44x0x1,x46x48x49x59xcx2f
’(x)? k +
Z x
fi
f (s)’(s) ds; fi? x? fl;
x2x70:
’(x)? ke
Rx
fi f(s)ds; fi? x? fl:
x70,x55
R(x) =
Z x
fi
f (s)’(s) ds;
x61
dR(x)
dx = f (x)’(x)? kf (x)+f (x)R(x):
x59xcx2fx77x78x1ex41x53x28x18x4ax18exp(Rxfi f (s) ds)x52x76
d
dx
exp
Z x
fi
f (s) ds
R(x)
kf (x)exp
Z x
fi
f (s) ds
;
x59xcx2fx77x78x75fix69xx29x1x76:
exp(?Rxfi f (s) ds)R(x)?Rxfi kf (t)exp
Rtfi f (s) ds
·
dt
=?kexp
Rtfi f (s) ds
·
jxfi = k[1?exp(?Rxfi f (s) ds)],
x59xcx2fx77x78x1ex41x53exp(Rxfi f (s) ds)x76R(x)? k[exp(Rxfi f (s) ds)?1].
x75x57’(x)? k +R(x)? kexp(Rxfi f (s) ds); fi? x? fl.
3,x6f (x;y)x41x39x39Gx3ax0x1x46x5ayx2fx7ax7bx59x2bx20,x2x70xbx40x11
x12y0 = f (x;y),y(x0) = y0 x20x3bx3cxdx2fx34x14x20.
x70,x39x7cx70xe,x18x3bx3cxdx59x34x14,x61x3x41xbx40x11x12x20x77x4exd:
’1 (x),’2 (x),x65x76x41x66x77x4exdx20x3ex1ex39x43[x0;b]x44x2fx59x47xcx20.
x55–(x) = [’2 (x)?’1 (x)]2,x61
d–(x)
dx = 2[’2 (x)?’1 (x)][f (x;’2 (x))?f (x;’1 (x))]? 0; x0? x? b
x8–(x)? 0; x0? x? b,x64x53–(x) · 0,x6x77x6x3fx40.x70x1d.
4.x77x6f (x;y)x41G = f(x;y)jfi? x? fl; y 2 Rgx44x0x1,x46x79x7a
yx48x49Lipschitzx73x74,( Lipschitzx16x18x31L),x2x70xbx40x11x12
y0 = f (x;y);y(x0) = y0 x20xdx41x7cx4ex39x43[fi;fl]x44x34x14x3x41.
x70,x41xdx20x3x41x34x14x1ax21xfx20x70x17x58x41x42x43x6ax0x52x70x17x41x8x12
x20x73x74x2a,xdx20x3x41x39x43x2fx41x7cx4ex39x43[fi;fl]x44x20,(x73x35x20x21xfx44
x31xdx20x5dx45x3x41x34x14x1ax21xf).x46x3fxfx0x39x43[x0;fl]x3cx3cx70x17,x8x70
14
x17x41x39x43 [x0;fl] x44x3x41x34x14xd,x41x70x17x58xfx28x29x27x47x3x44x20x70x17
x59x1ex20x48x2.
x6ax5e M = max
(x;y)2R
jf (x;y0)j,h = fl? x0,x61x27x47x3x44x20x70x17x14
x6e,x52x76x49x31x4ax37x4bx2bf’n (x)g,x36x2fx38x4cx20,x2cx46 j’n (x)j? jy0j +
M
eLh?1
·
=L,x5bx5fx49x31x4ax37x4bx2bx38x56x45.x53x2ax70x17x1ex47x3.
5.x6f (x;y)x41R2 x44x0x1,x5dx70,x5a8x0 2 R,xfx51jy0jx15x1x4d,xb
x40x11x12y0 =?y2?e2x￠f (x;y);y(x0) = y0 x20xdx37x52x4ex4fx69[x0;+1).
x70,x5ax7a8x0 2 R,x6bjy0j < ex0,x46x3fx70x17,x66x1fxbx40x11x12x20xdx20
x3bx3cxdy(x)x20x29x1x72x19x72x50x51x7ax77x72x19,y = §ex; (x? x?0)x52
x43x20x39x39Dx58.x66x6e,x3x4x4ex4fx21xfx52x5c,x3bx3cxdx14x68x52x53x24x3bx4ex4f
x69x36x29x53x2c,x39x7cx70xe,(x4x7ax41x = x0x1f,x29x1x72x19x51x7aDx58)x59x54,
x29x1x72x19x0x51x41x42x4ex1fx23x = x1 > x0 x1fx6x72x19y2 = e2x xbx31x73x5c,
x8x41x39x43[x0;x1)x44x29x1x72x19x51x7aDx3a,x57jy(x1)j = ex1,x4bx2f,x41x6a
(x1;y(x1))x44,x29x1x72x19x20x32x19x20x74x2cxcx7ax24,x0x2fx1e,x29x1x72x19x41
x6ax1;y(x1)x2fx75Dx22x3cx1bDx3ax20,x76x3fx40,x70x1d.
6.x6xbx40x11x12y0 = sin yx;y(x0) = y0 x20xdx31y = ’(x;x0;y0),x2x5d
@’
@x0 (x;x0;y0)x27
@’
@y0 (x;x0;y0)x3dx0 = 1;y0 = 0x1fx20x33x69x2f.
xd,@’@x
0
(x;x0;y0)x27 @’@y
0
(x;x0;y0)x3dx0 = 1;y0 = 0x1fx20x33x69x2fx1
x3ax2fx0x1x2x3
dz
dx =
cos
’(x;1;0)
x
·
x z
x1x3ax48x49xbx72x73x74z(1) = 0x27z(1) = 1x20xd,x64x53 @’@x
0
(x;1;0) = 0,
x67
@’
@y0 (x;1;0) = exp
2
4
Z x
1
cos
’(t;1;0)
t
·
t dt
3
5:
x4x7a’(x;1;0) = 0,x37x1bx44x2fx76 @’@y0 (x;1;0) = x.
7,x6f (x;y)x41Gx44x0x1x52x0,x2x70xbx40x11x12
y0 = f (x;y);y(x0) = y0 x20xdy = ’(x;x0;y0)x48x49x47xcx2f
@’
@x0 (x;x0;y0)+
@’
@y0 (x;x0;y0)f (x0;y0) · 0.
x70,x5bx31 @’@x
0
(x;x0;y0)x27 @’@y
0
(x;x0;y0)x24x2fx14x1dx19x1ax30x31x2x3
dz
dx?
@f
@y (x;’(x;x0;y0))z = 0x20xd,x64x53
z(x) = @’@x
0
(x;x0;y0)+ @’@y
0
(x;x0;y0)f (x0;y0)x2fx48x49xbx72x73x74
z(x0) = 0x20xd,x57x22z = 0x4ax2fx48x49xbx72x73x74z(x0) = 0x20xd,x4xd
x20x34x14x1a,z(x) · 0.
15
8,x5dxdx2ax2bx9x2x3
1) x = y?y02 +2y0
x32,x4y0 = p;y = x+p2?2p,x2ex45yx76
1+2(p?1)dp=dx?p = 0,x7cxfx76(p?1)(2dp=dx?1) = 0,x4p = 1,x76x3fxd
y = x?1,x42dp=dx = 1,x76p = x=2+c+1x56x3dxdx4e y = x+(x=2+c)2?1.
2) y = xy0 +
q
1+y02
x32,x2fClairautx2x3,x3dxdx31y = cx +p1+c2,x3fxdy =
p
1?x2.
x66x3fxdx2fx6exd,x5bx31Clairautx2x3x20x3fxdx0x2fx3dxdx20x63x6f,x75x57x2fx6e
xd,(x52x53x6fx70x36x2fx63x6f,x70x7ex3fxdx51x48x67x18x4fx2fx =?sint,y = cost,
t 2
…2; …2
,x61x71x3fxdx20x29x1x72x19x44x20x55x14x6a(?sint;cost)x38x3dxd
x58x6bc = tantx1fx20x29x1x72x19y = xtant+sectx41x66x6ax73x32).
3) y02 +xy0?y = 0
x32,x2fClairautx2x3,x3dxdx31y = cx+c2,x3fxdy =?x
2
4, x66x3fxdx2fx6exd
.
4) xy03?yy02?1 = 0
x32,x2fClairautx2x3,x3dxdx31y = cx?c?2,x3fxdy =?32 3
p
2x2,x66
x3fxdx2fx6exd.
5) y = 2x+y0? 13y03
x32,x3dxdx31
x =?12p2?2p?3ln[c(p?2)]; y =?p2?3p?6ln[c(p?2)]? 13p3,x3fxd
x31y = 2x? 23,x36x59x2fx6exd,x5bx31x3dxdx20x17x73x29x1x72x19(x41x32x19x74x2c
p =?1x20x6ax44)x27x3fxd(x32x19x74x2c= 2)x73x5cx57x59x73x32,x56x59x2fx6exd.
6) x?y = 49y02? 827y03
xd,x13x3cx67x18p = y0,x2x3x52x51x48x67x18x4fx2f
y0 = p; y = x? 49y02 + 827y03x6bx2ex45yx76x561? 29pp0 + 89p2p0?p = 0,x7cxf
x76(p?1)(8pp0=9?1) = 0,x48pp0=9?1 = 0,x84=9d(p2)=dx = 1,x29x1x76
p = §3px+c=2,x37x1byx20x33x69x2fx76x3dxdy = x+(x+c)[§px+c?1],
x4p?1 = 0,x37x1byx20x33x69x2fx76x3fxd,y = x?4=27.
9)x58x39Clairautx2x3x75x76x14x4ex53y = ’(x)x31x6exdx20x14x1dx2x3,x66
x5x77x6’ 2 C1[a;b],x46’0(x)x31xx20x78x79x7ax7bx4ax18.
xd,x55p = ’0(x),x5b’0(x)x31xx20x78x79x7ax7bx4ax18,x36x38x7cx4ax18,x6x31
x =?(p),x6x71x4ax18y = ’(x)x20x7dx7ex44x20x6ax20x32x19x2x3x31Clairautx2
x3y = px+f(p),x71x58p = dydxx61x52x46x41x32x6ax44x37x38f(p) = ’(x)?px =
’(?(p))?p?(p),xbx2dx6fx70y = ’(x)x2fx66x4eClairautx2x3x20x14x4ex6exd.
10,x7fx5c Riccati x2x3 y0 = cosx? (y? sinx)2 x38xd y = sinx,x18
x53 y = ’(x;x0;y0) x5ex4dx2x3x48x49xbx72x73x74 y(x0) = y0 x20xd,x2x5dx29
@’
@x0(x;0;1)x27
@’
@y0(x;0;1).
16
xd,x13x3cx44x20x5bx5cx4ax18u,x48x49y = sinx+ 1u,x37x1bx2x3x76 dudx = 1,
x29x1x76u = x+c,x75x57x4dRiccatix2x3x20x3dxdx31y = sinx+ 1x+c,x48x49
xbx72x73x74y(0) = 1x20x3fxdx31y = sinx+ 1x+1,x7ax2f,x64x5dx20x77x4ex1ax5e
x18x24x2fx19x1ax30x31x0x1x2x3 dzdx =? 2x+1zx20xd,x1x3ax48x49xbx72x73x74
z(0) =?
cosx?(y?sinx)2
·flfl
flx=0;y=1 = 0,x27z(0) = 1,x4x19x1ax2x3x20x3d
xdx2fz = c(x+1)2,x4xbx72x73x74,x1x3ax76c = 0x67c = 1,x56
@’
@x0(x;0;1) = 0,
@’
@y0(x;0;1) = (x+1)
2.
11*,x77x6x4ax18f(x;y)x41x39x39G ‰ R2x58x79x7ayx48x49x53Lx31Lpschitz
x16x18x20Lipschitzx73x74x6b ’(x)x27?(x)x31x2x3y0 = f(x;y)x41[a;b]x44x20x77
x4exdx6b x0 2 [a;b]x6bx2x70x4e
j’(x)(x)j? eLjx?x0jj’(x0)(x0)j x5a8x 2 [a;b]:
x70x17x4ex58x39x73x121.3x192x12x20x75x76x54x70.
12*,x5dx29x2ax2bRiccatix2x3xbx40x11x12
(1) dydx = y2 +?y +1; y(0) = 1; (2) dydx = y2 +?; y(0) = 0
x20xdy = ’(x;?)x79x7ax4dx67x18? x53x4fx2fx2x27x37x35x32x20x36xdx79x6bx2cx6x4d
x11x12x5cx36xdx5a?x20x10x58x18x35x32x3cx3cx15x4c.
xdx4e (1)x52x9x29x4dRiccatix2x3x38x3fxdyp =?1,x55y =?1+1=ux52
x7ex66Riccatix2x3x2bx31x5bx5cx4ax18ux20x19x1ax2x3du=dx = (2)u?1,
xdx76u = 1=(2)+cexp((2)x),x3x4xbx40x73x74y(0) = 1x76x69x2x3
x20x5cx36xdx31’(x;?) = 1+?(2)=(2exp((2)x)).
x55y = ’(x;?) = ’0 +?’1 +?2’2 +,::x6bx37x1bx0x1x2x3x6bx15x4c?x20
x1ex31x10x6bx76
’00 = ’20?1; ’(0) = 1;
’01 = 2’0’1 +’0 +1; ’1(0) = 0;
’02 = 2’0’2 +’21 +’1; ’2(0) = 0:
xdx76x4e ’0 = 1,’1(x) = exp(2x)?1,’2(x) = exp(2x)(exp(2x)?2x?1)=2.
(2)x2fx2ax1ax1x3ax2x3x6bx5cx36xdx2f
’(x;?) =
8
><
>:
p?tan(p?x); x3d 0;
pj?j(1?e2pj?jx)
1+e2
p
j?jx =
ptanh(px); x3d 0
x7ex53x44x19x14x2fx20tan(:)x42x10x58x18x35x32x8x76
’(x;?)? x?(1+ 13?x2 + 215(?x2)2 +::::
17
13*,x77x6x4ax18f(x;y)x27g(x;y)x41G ‰ R2x58x0x1x6bx46x5ax 2 [a;b)
x38 f(x;y)? g(x;y)x37x70x76 ’(x) x27?(x) x1x3ax31x2x3 y0 = f(x;y) x27
y0 = g(x;y) x41x39x43 [a;b) x44x48x49x1ex14xbx72x73x74 y(a) = y0 x20xdx6bx18
’(x) x6?(x) x58x39x12x38x14x4ex2fxbx72x11x12x41x39x43 [a;b) x44x20x34x14xdx6b
x2x70x41x39x43[a;b)x44x6b ’(x)(x).
x70x17,x18?(x) x2fxbx72x11x12x41x39x43 [a;b) x44x20x34x14xdx6bx4x12x56
’0(x) = f(x;’(x))? g(x;’(x)),x 2 [a;b),x42(x;y)x20x0x1x4ax18
F(x;y) =
(
g(x;y); x3dy? ’(x);
g(x;’(x)); x3dy < ’(x).
x6bxbx40x11x12y0 = F(x;y);y(a) = y0x20x3bx3cx3ax27xdx4e y = y(x);x 2 [a;T).
x59x70x17x41x39x43I · [a;min(b;T))x44x6b y(x)? ’(x).
x70x4ex18x59x54x6bx61x3x41x39x43(x1;x2) ‰ I,x41x5fx39x43x44y(x) < ’(x),x46
y(x1) = ’(x1),x4b’0(x)? g(x;’(x)) = F(x;y) = y0(x),x75x57’(x)? y(x),
x6x77x6x3fx40.
x5bx5fy = y(x)x2fx41x39x43Ix44xbx40x11x12y0 = g(x;y); y(a) = y0x6bx20
xd,x4xdx20x34x14x1ax6bx41x39x43I x44x6b y(x) ·?(x)x6bx3x4x7ay = y(x)x2f
x3bx3cx3ax27xdx6bx64x53T? b,x75x57x6bx41x39x43[a;b)x44?(x)? ’(x),x70x1d
x5ax7a’(x)x2fxbx72x11x12x41x39x43[a;b)x44x20x34x14xdx20x12x13x6bx52x53xe
x38x70x17x6bxfx51x65x44x7fx70x17x58x20?(x)x6’(x)x5cx50x6b gx6f x5cx50x6b?
x6?x5cx50x8x52.
14*,x77x6x4ax18f(y)x5ay 2 Rx0x1x6bx46x38f(0) = 0x6bx4bx3dy 6= 0x1f
x38f(y) 6= 0x37x2x70x4exbx40x11x12 dydx = f(y); y(0) = 0x38x34x14xdx20x15x51
x73x74x2fx4ex5a8a 6= 0;a 2 Rx6bx24x38 aR
0
ds
f(s) = 1.
x70x4ex37x51x1ax4ex18xbx40x11x12x38x34x14xdy = 0x6bx18x3x41a 6= 0,x65
aR
0
ds
f(s) = c 6= 1,x61x4x =
yR
a
ds
f(s) +c,(yx410x6ax52x43)x36x21x20x4ax18x2f
xbx40x11x12x20xd,x3fx40.
x15x1x1ax4ex18x73x74x48x62x57x34x14x1ax59x48x62x6bx61x5ax7ax42x4ea 6= 0,x3ex38
xdx =
yR
a
ds
f(s) +cx48x49xbx72x73x74x6bx8
aR
0
ds
f(s) = c 6= 1,x6x73x74x3fx40.
18
x26 x27 2.1
1,x5ax7ax2x3x1b~x0 = A(t)~x; ~x 2 Rn; t 2 I 4= [a;b];x20x55x14xd~x = ~x(t),
x70x17x5at? t0x59xcx2f
j~x(t)j?j~x(t0)jexp(
tZ
t0
(s)ds) 8t? t0 2 I
x48x62x6bx71x58j￠jx31j￠j2,?(t)x31x57x56Ax20x5ax44x9x1S(t) · (A(t)+A(t)T)=2
x20x30x2cx3fx5dx40.
x70x4ex2x3x1b~x0 = A(t)~xx77x78x74x41~xT x76
~xT~x0 = ~xTA(t)~x = ~xTS(t)~x(t)~xT~x
x8j~x(t)j0(t)j~x(t)j,x8
2
4j~x(t)jexp?(
tZ
t0
(s)ds)
3
5
0
0;
x29x1x2cx4xbx72x73x74x76x64x51x70x17x20x59xcx2f.
2,x70x17x55x1ax0x1x2x3x20Cauchyx11x12
~x0 = ~f (t;~x); ~x(t0) = ~x0;
x20xdx6x55x1aVolterrax29x1x2x3
~x(t) = ~x0 +
Z t
t0
~f (s;~x(s)) ds
x20x0x1xdx20xcx1bx1a.
x70,x6~x = ~’(t)x2fCauchyx11x12x20xd,x8
d~’(t)
dt =
~f (t; ~’(t)); ~’(t0) = ~x0;
x2x3x77x78x75t0 x69tx29x1,x76
~’(t) = ~’(t0)+
Z t
t0
~f (s; ~’(s)) ds
x8 ~’(t)x2fVolterrax2x3x20xd.
x7cx52,x18~’(t)x2fVolterrax2x3x20xd,x8
~’(t) = ~x0 +
Z t
t0
~f (s; ~’(s)) ds
19
x61 ~’(t0) = ~x0,x46x5b ~’(t)x0x1,x29x1x2ftx20x52x0x4ax18,x75x57 ~’(t)x52x0,
x5ax77x78x5dx5ex76
d~’(t)
dt =
~f (t; ~’(t));
x8x36x2fCauchyx11x12x20xd,x70x1d.
3,x5ax55x6cx20nx1dx16x2x56A,x70x17x57x56x58x18
I +A+ 12!A2 +￠￠￠+ 1n!An +￠￠￠
x20x59x5ax1a,x71x58I x31nx1dx7ax51x2x56.
x70,x58x39jIj = 1;jAmj?jAjm,x61x58x18
1+jAj+ 12!jAj2 +￠￠￠+ 1m!jAjm +￠￠￠ = ejAj
x31x12x58x58x18x20x5bx58x18,x75x57x4dx58x18x59x5a.
4,x7ex2ax2bx2x3x1bx51x48x14x1dx2x3x1b~x0 = A~x+ ~f(t)x20x4fx2f,x8x5dx29
Ax27 ~f(t),8
>><
>>:
dI1
dt +a1 (I1?I2) =
1
R1
de
dt(t);
d2I2
dt2 +a2
dI2
dt +a3I2?a4I1 = 0:
xd,x55x1 = I1,x2 = I2,x3 = dI2dt,x5e~x = (x1;x2;x3)T,x61x2x3x1bx52
x51x48~x0 = A~x+ ~f(t)x20x4fx2f,x71x58A =
2
64?a1 a1 00 0 1
a4?a3?a2
3
75,
f = [1=R1de=dt;0;0]T.
5,x2x39Picardx49x31x4ax37xe,x5dx29xbx40x11x12
d~x
dt =
"
0 4
1 0
#
~x; ~x(0) =
"
0
1
#
x20x193x31x37x38xd ~’3.
x32:
~’3 =
2
4 4t?
8
3t
3
1?2t2
3
5,(x7d,x5cx36xdx31~x =
"
2sin2t
cos2t
#
)
6,x6Ax31nx1dx16x2x56,I x31nx1dx7ax51x56,x46x3fx44?x20nx31x37x18
x2x3det(?I?A) = 0x31x2x3x1b~x0 = A~xx20x3fx5dx2x3,x2x51x29nx1dx16
x17x18x2x3x(n) +a1x(n?1) +￠￠￠+an?1x0 +anx = 0x20x3fx5dx2x3.
x32,x3fx5dx2x3x31?n +a1?n?1 +￠￠￠+an?1?+an = 0.
7,x6A =
"
0 1
1 0
#;x5ax7ax2x3x1b~x0 = A~x,x6fx70:
20
1) x38xd~u(t) =
"
cost
sint
#
x27~v(t) =
"
sint
cost
#;
2) x5a8c1;c2,x36x48x49xbx72x73x74~x(0) = (c1;c2)T x20xdx31
~x(t) = c1~u(t)+c2~v(t):
xd,x5e.
8,x7ex2ax2bx1ex1dx2x3x2bx48x1ex5fx14x1dx2x3x1bx53x71x58a;?;!;"x25x31x67
x18x2:
1x2d2ud 2 +u = a+"u2; (xfx60x3cx61x20 Einstein x2x3):
2x2d2xdt2 +?(x2?1)dxdt +x = 0; (Van der Pol x2x3):
3) d
2x
dt2 +!
2x+ax3 = 0; (Duffing x2x3):
4) d
2x
dt2
"
1?
dx
dt
2# dx
dt +x = 0; (Rayleigh x62x63).
xd:1,dud = v; dvd = a?u+"u2 ;
2,dxdt = y(x
3
3?x);
dy
dt =?x;(Li′enardx2ax50)
3,dxdt = y; dydt =?!2x?ax3 ;
4,dxdt = y; dydt =?x+?(1?y2)y.,
9,x64x59x58x39Li′enardx2ax50x65x1ex1dx66x19x1ax2x3
d2x
dt2 +x
2dx
dt +x
2 = 0
x2bx48x79x7a(x;y)x20x1ex5fx14x1dx2x3x1b,x54x7fx67x2ax48x53yx31x5ax2ax1ax3ex53xx31
x5bx5cx4ax18x20Bernoullix2x3,x2cx5dx29x66Bernoullix2x3x20x3dxd.
xd,x42Li′enardx2ax50,F(x) =
Z x
0
x2dx = x
3
3 ; y =
dx
dt + F(x),x76x1ex5f
x14x1dx2x3x1bx0 = y? x33 ; y0 =?x2,x36x2fx53yx31x5ax2ax1ax3ex53xx31x5bx5c
x4ax18x20Bernoullix2x3,x3cx57x2bx31x19x1ax2x3 d(x3)dy = x3?3y,x5dx76x3dxd
x3 = 3y +3+cey
21
x26 x27 2.2
x19x1ax23x30x31x0x1x2x3x1bx20x16x18x2ax2dx2ex2f,x5ax7ax19x1ax23x30x31x0x1
x2x3x1b d~xdt = A(t)~x+ ~f(t),x71x58A(t)x2fnx1dx0x1x4ax18x57x56,~f(t)x2fn
x5fx0x1x20x4ax18x55x1a,x6x5ax35x20x30x31x2x3x1bx20x7xdx57x56x31Φ(t),x61x23
x30x31x2x3x1bx20x3dxdx31~x(t) = Φ(t)
~c+
Z t
t0
Φ?1(s)~f(s)ds
,x71x58Φ?1(t)
x31Φ(t)x20x68x56,~cx31nx5fx55x56x16x18x55x1a.
1,x2x6fx70
"
t2 t
2t 1
#
x31x30x31x2x3x1b~x0 =
2
4 0 1
2t2 2t
3
5~xx41t > 0x44
x20x7xdx56,x2cx5dxdx23x30x31x2x3x1bx20xbx40x11x12
8>
>>>>
<
>>>>
>:
~x0 =
2
4 0 1
2t2 2t
3
5~x+
"
t
1
#;
~x(1) =
"
1
1
#
x32,x6fx70x5e,x39x16x18x2ax2dx2ex2fx76xdx31~x(t) =?t2;t￠T.
2,x6nx1dx2x56Φ(t)x20x2bx55x1ax41[a;b]x44x19x1ax29x79,x69x29x41x33x34
x73x74x2ax6ax3x41x30x31x2x3x1b~x0 = A(t)~x,(x71x58A(t)x41[a;b]x44x0x1)x65
x76Φ(t)x31x36x20x7xdx56x35x18x66x6ex20x2x3x1bx3x41,x2fx30x34x14x35
x32,x3dx46x16x3dnx1dx2x56Φ(t)x20x3cx2bx2fx41[a;b]x44x47x59x31x24x46x0
x1x52x0x1f,x6ax3x41x30x31x2x3x1b~x0 = A(t)~x,x65x76Φ(t)x31x36x20x7xdx56,
x57x46x66x6ex20x2x3x1bx2fx34x14x20(x5bx31x66x1fA(t) = Φ0(t)Φ?1 (t)).
3,x6A(t)x31x41[a;b]x44x0x1x20nx1dx1fx2x56,Φ(t)x31x2x3x1b
~x0 = A(t)~xx20x7xdx56,x57 ~’(t)x31x36x20x14x4exd,x2x70:
(i)x5ax7ax2x3x1b~y0 =?AT (t)~yx20x55x14xd~y = ~?(t)x37x38
~?T (t) ~’(t) =x16x18:
(ii) Ψ(t)x31x2x3x1b~y0 =?AT(t)~y x20x7xdx56x20x15x51x73x74x31x3x41x23
x6bx20x16x2x56C,x65x76ΨT (t)Φ(t) = C.
x70,(i)x5bx31h~?T ~’
i0
= ~?0T ~’+ ~?T ~’0 = =?
AT ~?
·T
~’+ ~?TA~’ = 0.
x64x53~?T (t) ~’(t) =x16x18.
(ii)x37x51x1a,x6Ψ(t)x31x2x3x1b~y0 =?AT (t)~yx20x7xdx56,x61Ψ(t)x20
x55x14x2bx55x1a ~?i (t)x2fx2x3x1b~y0 =?AT (t)~yx20xd,x57Φ(t)x20x55x14x2bx55
x1a ~’j (t)x2fx2x3x1b~x0 = A(t)~xx20xd,x4(i),~?Ti (t) ~’j (t) = cij x2fx16x18:
x8ΨT (t)Φ(t) = C =
cij
·x2fx16x2x56
.
x15x1x1a,x6ΨT (t)Φ(t) = Cx2fx23x6bx20x16x2x56,x4x7ax7xdx56Φx2fx23
x6bx20,x61ΨT = CΦ?1 x4ax23x6b,x77x78x79x7atx5dx5ex76:
Ψ0T = C
Φ?1
·0; (1)
22
x3x5ax47xcx2fΦ?1Φ = I x77x78x79x7atx5dx5ex76,?Φ?1￠0Φ+Φ?1Φ0,x75x57x76
Φ?1
·0
=?Φ?1Φ0Φ?1; (2)
x37x1b(1)x2fx76Ψ0T = C(Φ?1)0 =?CΦ?1Φ0Φ?1 =?CΦ?1AΦΦ?1 =?ΨTA,
x8Ψ0T =?ΨTA,x77x78x6bx67x6cx76Ψ0 =?ATΨ,x8Ψx2f~y0 =?AT(t)~yx20
x7xdx56.x70x1d.
4,x6Ax31nx1dx16x2x56,Φ(t)x31x2x3x1b~x0 = A~xx20x7x35x7xdx56,x2
x70,x5a8t;t0 2 Rx38Φ(t)Φ?1 (t0) = Φ(t?t0).
x70,x4x7aΦ(t)Φ?1 (t0) = Φ(t?t0)x20x77x78x24x2fx7xdx56,x46x41t = t0
x1fx73xc.x4x0x1x2x3x57x56xdx20x34x14x1ax76x5c,x66x77x4ex7xdx56x47xc,x70
x1d.
5,x6A(t)x27 ~f(t)x1x3ax31x41[a;b]x44x0x1x20nx1dx2x56x27nx5fx55x1a,
x70x17x2x3x1b~x0 = A(t)~x+ ~f(t); ~f(t) 6· 0,x3x41x46x30x6dx3x41n+1x4ex19
x1ax29x79xd.
x70,x6Φ(t)x31x5ax35x20x30x31x2x3x1bx20x7xdx56,~’j(t)x2fΨ(t)x20x19j
x2b,~’(t)x31x23x30x31x2x3x1bx20x55x14x3fxd,x61 ~?0(t) = ~’(t),~?j(t) = ~’(t) +
~’j(t),j = 1;2;:::;nx2fx23x30x31x2x3x1bx20n+1x4exd,x16x70x36x3fx19x1ax29
x79,x39x7cx70xe.x18x36x3fx19x1ax73x79,x61x3x41n + 1x4ex59x5dx31x24x20x16x18cj,
j = 0;1;2;:::;n,x65x76cj ~?j(t) = 0,x8Σn0cj~’(t) =?Σn1cj~’j(t),x4 ~’j(t),
j = 0;1;2;:::;nx20x19x1ax29x79x1a,x52x5cΣn0cj 6= 0,x64x53 ~’(t)x2fx5ax35x20x30
x31x2x3x1bx20xd,x6x77x6 ~’(t)x31x23x30x31x2x3x1bx20x3fxdx3fx40.
x3x70x23x30x31x2x3x1bx30x6dx3x41n+1x4ex19x1ax29x79xd,x39x7cx70xe,x18
~?j(t),j = 0;1;2;:::;n+1x2fx23x30x31x2x3x1bx20n+2x4ex19x1ax29x79xd,x61
x52x46 ~’j(t) = ~?j(t)? ~?0(t),j = 1;2;￠￠￠;n+1x2fx30x31x2x3x1bx20n+1x4e
x19x1ax29x79xd,x3fx40.
6,x2x70x19x1ax23x30x31x2x3x1bxdx20x6ex6fx3bxf,x6~x1 (t)x27~x2 (t)x1x3a
x2fx2x3x1b~x0 = A(t)~x+~f1 (t)x27~x0 = A(t)~x+~f2 (t)x20xd,x61~x1 (t)+~x2 (t)
x2fx2x3x1b~x0 = A(t)~x+ ~f1 (t)+ ~f2 (t)x20xd.
x70x17,x68x70x37x1bx6fx70x8x76.
7,x6cx21x2x3x1b~x0 = A~x+ ~f (t),x71x58
A =
"
2 1
0 2
#; ~f (t) =
"
sint
cost
#
:
(i)x2x6fx70Φ(t) =
"
e2t te2t
0 e2t
#
x2f~x0 = A~xx20x7xdx56:
(ii)x2x5dx2x3x1b~x0 = A~x + ~f (t)x48x49xbx72x73x74~x(0) = (1;?1)T x20
xd.
x32:
(i)x6fx70x5e.
(ii)xdx31:
~’(t) =
"?27
25?
3
5t
·
e2t? 225 cost? 1425 sint
35e2t? 25 cost+ 15 sint
#
:
23
8.x2x5dxbx40x11x12~x0 = A~x+ ~f (~x); ~x(0) = (1;?1)T x20xd ~’(t),x71x58
A =
"
2 1
0 2
#; ~f (t) =
"
0
e2t
#
:
x32,xdx31,~’(t) = e2t
1?t+ t
2
2 ;?1+t
!T
.
9.x6cx21x2x3x1b~x0 = A~x+~f (t),x71x58A =
"
0 1
7?8
#; ~f (t)x41[0;+1)
x44x0x1,x2x68x70x6fx70x4fx58x39x16x18x2ax2dx2ex2fx70x17:
(i)Φ(t) =
"
e?t?e?7t
e?t 7e?7t
#
x31~x0 = A~xx20x7xdx56:
(ii)x18 ~f (t) x41 [0;+1) x44x38x4c,x61 ~x0 = A~x + ~f (~x) x20x17x14x4exdx41
[0;+1)x44x38x4c:
(iii)x18x3dt ! +1x1f,~f (t) ! 0,x61~x0 = A~x + ~f (t)x20x17x14x4exd ~’(t)
x48x49:x3dt ! +1x1f,~’(t) ! 0.
x70:
(i)x68x70x6fx70x52x76,Φ0(t) =
"
e?t 7e?7t
e?t?49e?7t
#
,
x57,AΦ0(t) =
"
0 1
7?8
#"
e?t?e?7t
e?t 7e?7t
#
=
"
e?t 7e?7t
e?t?49e?7t
#
,
x64x53Φ0(t) = AΦ(t),x8Φ(t)x31~x0 = A~xx20x7xdx56:
(ii)x6j~f (t)j = max(jfjj)? M,x4x16x18x2ax2dx2ex2f,xdx52x33x34x31
~’(t) = Φ(t)~c+
Z t
0
Φ(t?s) ~f (s) ds
x61
j~’(t)j?jΦ(t)jj~cj+
Z t
0
jΦ(t?s)jj~f (s)jds
x71x72jΦ(t)j = 7e?7t +e?t; j~f (t)j? M,x52x76
j~’(t)j7e?7t +e?t￠j~cj+Rt0[7e?7(t?s) +e?(t?s)]M ds
=?7e?7t +e?t￠j~cj+M
h
e?7(t?s) +e?(t?s)
i
jt0? 8j~cj+2M.
(iii)x6j~f (t)j? M,x46x55x56x6cx21x20? > 0,x3x41T > 0,x65x76t > T
x1f,j~f (t)j <?,x75x57x3dt > T x1f
j~’(t)j?
h
7e?7t +e?t]j~cj+RT0 [7e?7(t?s) +e?(t?s)
i
M ds+
+RtT
h
7e?7(t?s) +e?(t?s)
i
ds =
= ￡7e?7t +e?t?j~cj+M
h
e?7(t?s) +e?(t?s)
iflfl
flT0 +?
h
e?7(t?s) +e?(t?s)
iflfl
tT
￡7e?7t +e?t?j~cj+M
h
e?7(t?T) +e?(t?T)
i
+2?:
24
x64x53x3dt(> T)x15x1x2cx1f,j’(t)j < 4?,x4?x20x55x56x1ax76,x3dt ! +1
x1f,
~’(t) ! 0.
10,x6nx1dx2x56A(t)x41[a;b]x44x0x1,Φ(t)x31~x0 = A(t)~xx20x14x4ex7
xdx56,nx5fx4ax18x55x1a ~f (t;~x)x79x7at 2 [a;b];~x 2 Rn x0x1,t0 2 [a;b],x2
x70xbx40x11x12
~x0 = A(t)~x+ ~f (t;~x);~x(t0) = ~x0
x20xdx27x29x1x2x3
~x(t) = Φ(t)Φ?1 (t0)~x0 +
Z t
t0
Φ(t)Φ?1 (s) ~f (s;~x(s)) ds
x20x0x1xdx73x1e.
x70,x77x4exdx41t = t0 x1fx73xc,x7ex29x1x2x3x77x78x79x7atx5dx5ex76:
~x0(t) = Φ0(t)Φ?1 (t0)~x0 +Rt0 Φ0(t)Φ?1 (s) ~f (s;~x(s)) ds+ ~f (t;~x(t))
= A(t)
h
Φ(t)Φ?1 (t0)~x0 +Rt0 Φ(t)Φ?1 (s) ~f (s;~x(s)) ds
i
+ ~f (t;~x(t))
= A(t)~x(t)+ ~f (t;~x(t)).
x8x29x1x2x3x20x0x1xdx2fx0x1x2x3xbx40x11x12x20xd.
x7cx52,x6~x(t)x2fxbx40x11x12x20xd,x7dx56x69Φ0(t) = A(t)Φ(t)x53x67x8
x73x123xdx32x52x2ex2f(2),?Φ?1￠0 =?Φ?1Φ0Φ?1.x7ex0x1x2x3x77x78x74x41x53
x57x56Φ?1 (t),x76
Φ?1 (t)~x0(t) = Φ?1 (t)A(t)~x(t)+Φ?1 (t) ~f (t;~x(t))
= Φ?1 (t)Φ0(t)Φ?1 (t)~x(t)+Φ?1 (t) ~f (t;~x(t))
=Φ?1￠0~x(t)+Φ?1 (t) ~f (t;~x(t)):
x8￡Φ?1 (t)~x(t)?0 = Φ?1 (t) ~f (t;~x(t)).
x77x78x79x7atx750x69tx29x1x76
Φ?1 (t)~x(t) = Φ?1 (t0)~x(t0)+Rtt0 Φ?1 (s) ~f (s;~x(s)) ds,
x77x78x74x41Φ(t)x8x76x5c~x(t)x2fx29x1x2x3x20xd.
11*x3ex58x39x78x40x11x12x20x75x76x70x17xbx40x11x12
~x0 = A(t)~x+ ~f(t); ~x(a) = ~·
xdx20x3x41x34x14x1ax21xf,x53x67x77x5exdx20x33x69x2fx53x16x18x2ax2dx2ex2fx532–27x2
x2,x71x58 nx1dx57x56A(t)x27nx5fx55x1a~f(t)x24x41[a;b]x44x0x1,~·x31x16x55x1a.
xd,x6 Φ(t) x2fx30x31x2x3x1bx20x7xdx56,x66x1f C = I,D = 0,x57x56
B = Φ(a)x23x6b,x64x53x30x31x78x40x11x12xfx38x24xd,x75x57x78x40x11x12x38x34x14
xd,Greenx4ax18x31
G(t;s) =
(
0; x3d a? t < sx1f:
Φ(t)Φ?1(s); x3d s < t? bx1f.
x75x57x76x16x18x2ax2dx2ex2f.
25
12*x3ex58x39x79x78x4ax18x5dxdx78x40x11x12x532–29x2,x71x58a = 0;b = 1;~· = 0,
x57
A(t) =
"
0 1
0 0
#; C =
"
1 0
0 0
#; D =
"
0 0
1 0
#; ~f(t) =
"
t
1
#
:
xd,x6bΦ(t) =
"
1 t
0 1
#
,x61B =
"
1 0
1 1
#
,B?1 =
"
1 0
1 1
#
,Green
x4ax18x31
G(t;s) =
(
Φ(t)B?1DΦ(1)Φ?1(s); x3d 0? t < sx1f:
Φ(t)B?1CΦ(0)Φ?1(s); x3d s < t? 1x1f.
xdx31~x(t) = R1t (?t;?1)Tds = (t(t?1);t?1)T.
x7d,x1fx20x44x8x12x4ax52x53x59x5dx29x2x3x20x3dxd~x(t) = Φ(t)~c + (t2;t)T,
x54x7fx4x78x40x73x74x54x7x21x16x18x24x1a~c.
26
x26 x27 2.3
x5ax7ax55x14?x20kx31x6dx79x2fp(?,x52x53x7ee?t x51x48x70x2ax4fx2f:
e?t = p(?)q(?;t)+c0(t)+c1(t)?+c2(t)?2 +￠￠￠+ck?1(t)?k?1
x71x58q(?;t)x2f?x6tx20x4ax18,x3dx7fx5cp(?)x20kx4ex71(x63x40x43x71)x1f,x52
x53x39x62x21x17x18xex36x21 kx4ex5bx5cx4ax18cj(t),j = 0;1;￠￠￠;k?1,x5ax7an
x1dx1fx16x18x2x56A,x6bp(?)x31Ax20x55x14x2bx24x6dx79x2f,(x8x65x76p(A) = 0
x20x6dx79x2fx6bx7ax70Ax20x3fx5dx6dx79x2fx2fj?I?Ajx2fAx20x2bx24x6dx79x2fx6b A
x20x30x4dx6dx79x2fx2fAx20x31x18x30x4dx20x2bx24x6dx79x2f),x7ax2fx4x2bx24x6dx79x2f
x20x1ax7bx52x46:
exp(At) = c0(t)I +c1(t)A+c2(t)A2 +￠￠￠+ck?1(t)Ak?1
x63x54x6bp(?)x31Ax20x30x4dx6dx79x2fx1fx7cx60x1ax30x4d.
x5bx5fx76x69x57x56x28x18exp(At)x20x7cx60x2xex70x2a,x6nx1dx1fx16x18x2
x56Ax20x30x4dx6dx79x2fx31?x20mx31x6dx79x2fpm(?),x6pm(?)x20x71x31?1,
2,￠￠￠,?s,x71x58x71?j x20x43x18x31kj,j = 1;2;￠￠￠s.
k1+k2+￠￠￠+ks = m,x4x19x1ax37x18x58x30x4dx6dx79x2fx20x21x45,x38,pm(A) = 0,
x52x6
exp(At) = c0(t)I +c1(t)A+c2(t)A2 +￠￠￠+cm?1(t)Am?1
x71x58mx4etx20x1fx4ax18c1(t),c2(t),￠￠￠,cm(t)x4x2ax2bmx4ex1fx19x1ax2x3
x7x21:
x5ax17x4ex43x18x31kj x20x1fx71?j,x51x29x2x3
c0 +c1?j +c2?2j +￠￠￠+cm?1?m?1j = e?jt
x3dkj > 1x1f,x4fx2fx48x5ax2x3x77x78x79x7a?j x7dx31x5dx5ekj?1x31,x3ex76kj
x4ex1fx2x3:
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:
x70x76?j x2fx7ex71,x61x52x7ex44x7fx2x3x1x29x1fx9x27x7ex9,x0x762kj x4ex1fx2
x3,x66x1f,x5ax7a?j x20x3ex0x7ex71x0x59x39x3x1x62x2x3x1(x5bx31x5ax7a?j x20
x3ex0x7ex71x1x62x202kj x4ex1fx2x3x6x5ax7a?j x1x62x202kj x4ex1fx2x3x2fx73
x1ex20),x66x6ex14x3ex38mx4ex1fx19x1ax2x3,x52x5dx29mx4ex1fx4ax18c0(t); c1(t),
￠￠￠,cm?1(t).
x2x3 d~xdt = A~xx48x49xbx72x73x74~x(0) = ~x0 x20xdx31~x = exp(At)~x0 =
c0(t)~x0 +c1(t)A~x0 +c2(t)A2~x0 +￠￠￠+cm?1(t)Am?1~x0 x18~x0 x36x2x2fAx20
x5ax35x7ax3fx5dx40?x20x3fx5dx55x1ax1f,x8A~x0 =?~x0,x61x4x21x45x52x76:
~x = exp(At)~x0 = e?t~x0:
27
x3dx3fx5dx6dx79x2fxfx38x14x4ex1fx71? = ax1f(x66x1fx30x4dx6dx79x2fx31(
a)m,m? n)x38x3x4x20x60xe:
exp(At) = exp(atI +(A?aI)t) = exp(atI)exp((A?aI)t) =
= exp(at)
h
I +(A?aI)t+ 12! (A?aI)2 t2 +￠￠￠+ 1(m?1)! (A?aI)m?1 tm?1
i
.
x30x4dx6dx79x2fx20x5dxe,? =?j x2fnx1dx1fx16x18x56Ax20x30x4dx6dx79x2fpm
x20x71x3dx46x16x3d? =?jx2fAx20x3fx5dx6dx79x2fx20x71,x5ax7a? =?j x2fx3fx5d
x6dx79x2fx20x7ax71x1f,x36x4ax2fx30x4dx6dx79x2fx20x7ax71,x3d? =?j x2fx3fx5dx6dx79
x2fx20k > 1x43x71x1f,x30x4dx6dx79x2fpm x20x71? =?j x20x43x18kj x2fAx20
Jordanx7x35x56x58x79x7ax3fx5dx40?j x20x30x2cx20Jordanx64x5x20x1dx18,x5bx5f
kj = minfrjrank(AjI)r = n?kg;
x71x58x5ex6rankBx33x34x57x56Bx20x7.
x7d,x53x44x2xex4ax52x53x39x54x5dx57x56x5ax18lnAt,x57x56x36x21 sinAt,x57
x56x3bx21 cosAt,xfx51x65e?jtx73x35x48x6ax31ln(?jt),sin(?jt),cos(?jt)x8x52.
x7a,x6 A =
"
a b
c d
#
,x3fx5dx40? = fi § fl,x71x58 fi = a+b2,fl2 =
bc+ (a?d)24 =?detB,x5bx31
exp(At) = exp(fitI)exp((A?fiI)t) = efit exp((A?fiI)t))
x7ax2fx2bx31x5dB = A?fiI x20x57x56x28x18,Bx20x3fx5dx40x31§fl,
exp(Bt) = c0(t)I +c1(t)B;
x71x58c0(t),c1(t)x48x49x2ax2bx2x3:
c0(t)+flc1(t) = eflt;
c0(t)?flc1(t) = e?flt:
x3dfl 6= 0x1f,xdx76
C0(t) = cosh(flt); c1(t) = sinh(flt)fl,
x7ax2fx76
exp(At) = efit
2
4 cosh(flt)+
a?d
2
sinh(flt)
fl b
sinh(flt)
fl
csinh(flt)fl cosh(flt)+ d?a2 sinh(flt)fl
3
5 (1):
x3d fl = !i 6= 0 x2fx8x7ex18x1f,x5bx31 cosh(!it) = cos(!t),sinh(!it) =
isin(!t),x64x53
exp(At) = efit
"
cos(!t)+ a?d2 sin(!t)! bsin(!t)!
csinh(!t)! cos(!t)+ d?a2 sin(!t)!
#
(2):
28
x3dfl = 0 (x2fBx20x64x38x20x71),x64x53
exp(At) = efit exp(Bt) = efit(I +tB)
x8
exp(At) = efit
"
1+ a?d2 t bt
ct 1+ d?a2 t
#
(3):
x7dx56x2ex2f(3)x52x53x9x48x2ex2f(1)x58x3dfl x23x240x1fx4fx2ex2f(2)x58x3d!
x23x240x1fx20xax2a.
x16x17x18x23x30x31x2x3x1bx20x16x18x2ax2dx2ex2f,x5ax7ax16x17x18x23x30x31x0x1
x2x3x1b
d~x
dt = A~x +
~f(t),x71x58Ax2fnx1dx1fx16x18x57x56,~f(t)x2fnx5fx0x1x20x4ax18
x55x1a,x23x30x31x2x3x1bx20x3dxdx31
~x(t) = exp(At)~c+
Z t
t0
exp(A(t?s))~f(s)ds,x71x58cx31nx5fx55x56x16x18x55x1a.
xbx40x11x12 d~xdt = A~x+ ~f(t),~x(0) = ~x0 x20xdx31:
~x(t) = exp(At)~x0 +
Z t
0
exp(A(t?s))~f(s)ds.
x5dx16x17x18x23x30x31x2x3x1bx20x3fxdx20x60x64x2xe,x5eD = ddt,Dk = d
k
dtk.x6
~y(t)x2fnx1dx16x17x18x0x1x2x3det(DI?A)[~y] = L(D)[~y] = ~f(t)x20x3fxd,
(L(?)x2fAx20x3fx5dx6dx79x2f)(x66x3fxdx20x5dxex46x73x122.4)x61ˉ~x(t) = (DI?
A)?~y(t)x2fx23x30x31x2x3x1bx20x3fxd,x71x58(DI?A)?x2fx60x64x57x56DI?A
x20xbxcx57x56.
1,x7cx60x2ax2bx2x56Ax20exp(At).
1)
"
2 1
4?1
#
,2)
"
1?1
4?3
#
,3)
"
1?3
3 1
#
,4)
"
1?5
1 1
#
.
5)
2
64 2?1?12?1?2
1 1 2
3
75,6)
2
64?2 1?21?2 2
3?3 5
3
75,7)
2
64 1?1 11 1?1
0?1 2
3
75.
8)
2
64 2?1 11 2?1
1?1 2
3
75,9)
2
64 2?1 21 0 2
2 1?1
3
75,10)
2
64 0 c?b?c 0 a
b?a 0
3
75.
xd,1)x3fx5dx6dx79x2f(?+2)(3)x20x71x24x2fx7ax71,x36x0x2fx30x4dx6dx79
x2f,xdx2x3x1b
c0?2c1 = e?2t; c0 +3c1 = e3t:
x76c1 = 15
e3t?e?2t
·; c0 = 15
2e3t +3e?2t
·
:
x64x53exp(At) = c0I + c1A = e3t
2
5I +
1
5A
+ e?2t
3
5I?
1
5A
,x7cxf
x76
exp(At) = 15
"
4e3t +e?2t e3t?e?2t
4?e3t?e?2t￠ e3t +4e?2t
#
:
29
xd,2)x3fx5dx6dx79x2fx2f(?+1)2.
exp(At) = exp(?tI +(A+I)t) =
= exp(?tI)exp((A+I)t) = exp(?t)[I +(A+I)t],x76
exp(At) = e?t
"
1+2t?t
4t 1?2t
#
:
xd,3)xdxe1,x3fx5dx2x3x31?2?2?+10 = 0,x3fx5dx40x2fx14x5ax3ex0xd
x71? = 1§3i,xdx2x3c0 +(1+3i)c1 = e(1+3i)t x55x2x3x77x78x1fx9x6x7ex9
x1x3ax73xc,x76x2x3x1bc0 +c1 = et cos(3t); 3c1 = et sin(3t),xdx29c0,c1x7f
x37x1bexp(At) = c0I +c1Ax7cxfx76
exp(At) = et
"
cos3t?sin3t
sin3t cos3t
#
:
xdxe2,x5bx31A = I +3J,x71x58
J =
"
0?1
1 0
#
.
x4x7aI x6J x52x5cx50,x7ax2fexp(At) = exp(It)exp(3Jt) = et exp(3Jt),
x3x4x7a J2 =?I,x71x72x57x56x28x18x20x21x45x52x5c exp(3Jt) = cos(3t)I +
sin(3t)J,x75x57x62x76exp(At) = et(cos(3t)I +sin(3t)J).
(x7d:x5ax7ax14x15x20A =
"
a?b
b a
#
,xex38x52x76exp(At) = eat (cos(bt)I +sin(bt)J).
xdxexd,xax2ex2f(2)x20x2x5dxex4c.
x32,4x2x3fx5dx6dx79x2fx31?2 +4 = 0,x4(2)x76
exp(At) =
2
64 cos2t?
1
2 sin2t?
5
2 sin2t1
2 sin2t cos2t+
1
2 sin2t
3
75:
xd,5)x3fx5dx6dx79x2fx2f(1)3,A?Ix20x7x2f1,x64x53x57x56Ax20Jordan
x7x35x56x58x79x7ax3fx5dx401x20x30x2cJordanx64x5x20x1dx18x2f2,x75x57x30x4dx6d
x79x2fx2f(1)2,x64x53exp(At) = exp(tI +(A?I)t) =
= exp(tI)exp((A?I)t) = exp(t)[I +(A?I)t],x76
exp(At) = et
2
64 1+t?t?t2t 1?2t?2t
t t 1+t
3
75:
xd,6) Ax20x3fx5dx6dx79x2fx2f(?+1)2(3),x4x7aA+Ix20x7x2f1,x64x53
x57x56Ax20Jordanx7x35x56x58x79x7ax3fx5dx40-1x20x30x2cJordanx64x5x20x1dx18
x2f1,x75x57x30x4dx6dx79x2fx2f(?+1)(3),x64x53x6exp(At) = c0(t)I+c1(t)A,
xdx2x3x1bc0?c1 = e?t,c0 + 3c1 = e3t,x76c1(t) = 14
e3t?e?t
·
,c0(t) =
30
1
4
e3t +3e?t
·
,x75x57
exp(At) = 14
2
64?e
3t +5e?t e3t?e?t 2(e?t?e3t)
e3t?e?t?e3t +5e?t 2(e3t?e?t)
3?e3t?e?t￠?3?e3t?e?t￠ 2(3e3t?e?t)
3
75:
x32,7)x8x12x30x4dx6dx79x2fx0x2fx3fx5dx6dx79x2f,(1)2(2),x5bexpAt =
et exp(A?I)t,x57B = A?Ix20x3fx5dx40x310 (x1ex43x71)x61,x3dx71x5dexpBt
x76
exp(At) = et
2
64 2+t?e
t 1?et 2et?(2+t)
t 1?t
1+t?et 1?et 2et?(1+t)
3
75:
x32,8)x8x12x30x4dx6dx79x2fx0x2fx3fx5dx6dx79x2f,(1)(2)(3),
exp(At) =
2
64 e
2t e2t?e3t e3t?e2t
e2t?et e2t et?e2t
e2t?et e2t?e3t et +e3t?e2t
3
75:
x32,9)
exp(At) =
2
66
4
cost+2sint?sint 2sint
cost?et +2sint et?sint 2sint
cost?et?3sint
2
et?cost+sint
2 cost?sint
3
77
5:
x32,10x2x3fx5dx2x3x31?(?2 + a2 + b2 + c2) = 0,x5efl = pa2 +b2 +c2,
xdx76,c0 = 1,c1 = 1fl sin(flt),c2 = 1fl2[1?cos(flt)],x75x57
exp(At) = 1fl2 [~’1(t) ~’2(t) ~’3(t)],x71x58
~’1(t) =
2
64 a
2 +(b2 +c2)cosflt
ab(1?cosflt)?flcsinflt
ca(1?cosflt)+flbsinflt
3
75,
~’2(t) =
2
64 ab(1?cosflt)+flcsinfltb2 +(c2 +a2)cosflt
bc(1?cosflt)?flasinflt
3
75,
~’3(t) =
2
64 ca(1?cosflt)?flbsinfltbc(1?cosflt)+flasinflt
c2 +(a2 +b2)cosflt
3
75.
2,x5dx29x2ax2bxbx40x11x12~x0 = A~x;~x(0) = ~x0 x20xd ~’(t).
1) A =
"
5 3
3?1
#;~x0 =
"
1
1
#
,2) A =
"
1?5
1 1
#;~x0 =
"
1
0
#
,
31
3) A =
2
64 3 1?1?1 2 1
1 1 1
3
75;~x
0 =
2
64 10
1
3
75,4) A =
2
64 2 1 00 2 4
1 0?1
3
75;~x
0 =
2
64 01
1
3
75,
5) A =
2
64 2 1 01 3?1
1 2 3
3
75;~x
0 =
2
64 11
0
3
75.
xd,1) x3fx5dx6dx79x2fx31 ( 2)2 = 0,~’(t) = e2t(I + (A? 2I)t)~x0 =
e2t (1+6t;1?6t)T
x32,2)x4x73x122.3x121.4 ~’(t) = exp(At)~x0 =
cos2t? 12 sin2t; 12 sin2t
T
xd,3) x5bx31 ~x0 x36x2x2f A x20x3fx5dx40 2 x5ax35x20x3fx5dx55x1a,x56~’(t) =
e2t (1;0;1)T
x32,4) ~’(t) =
8
9e
3t? 8
9?
5
3t;
1
9 +
10
3 t+
8
9e
3t; 7
9 +
2
9e
3t? 5
3t
T
x32,5) ~’(t) =?e3t sint+e2t;e3t sint+e3t cost;2e3t sint?e3t cost+e2t￠T
3,x5dx29x2ax2bxbx40x11x12~x0 = A~x+ ~f (t);~x(0) = ~x0 x20xd ~’(t).
1) A =
"
1 8
1 1
#; ~f (t) =
"
et
e?t
#;~x0 =
"
0
1
#
,
2) A =
"
1?5
1 1
#; ~f (t) =
"
1
t
#;~x0 =
"
1
1
#
,
3) A =
2
64 2?1?13?2?3
1 1 2
3
75; ~f (t) =
2
64 1t
2t
3
75;~x
0 =
2
64 10
1
3
75,
4)A =
2
64 1?1 11 1?1
2?1 0
3
75; ~f (t) =
2
64 11
1
3
75;~x
0 =
2
64 11
0
3
75,
5) A =
2
64 2?1 21 0 2
2 1?1
3
75; ~f (t) =
2
64 11
t
3
75;~x
0 =
2
64 01
1
3
75.
x32,1) Ax20x3fx5dx40x31§3,x6exp(At) = c0I +c1A,x71x58
c0 +3c1 = e3t; c0?3c1 = e?3t:
xdx29c0 = cosh(3t),c1 = 13 sinh(3t).
x3x5dx23x30x31x2x3x20xd,x59x5dx2x3det(DI?A)~y = ~f (t)x20x3fxd.
x8x5d?D2?9￠~y = ~f (t)x20x3fxd~yp (t) =?18
et;e?t
·T
,x7ax2f
~xp (t) = (DI?A)?~yp (t) =
e?t;?18et
T
x2fx3bx23x30x31x2x3x20x14x4ex3fxd.x71x58(DI?A)?x2fDI?Ax20xbxcx57x56:
(DI?A)? =
"
D?1 8
1 D+1
#;
32
x75x57x2x3x20x3dxdx31~x(t) = ~xp (t) + exp(At)~c,x57x48x49xbx72x73x74x20
xdx31
~x(t) = ~xp (t)+exp(At)(~x0?~xp(0))
x76x64x5dx20xdx31
~x(t) = ~xp (t)+c0(t)(~x0?~xp(0))+c1(t)A(~x0?~xp(0)) =
" 11
6 e
3t? 5
6e
3t?e?t
11
12e
3t + 5
24e
3t? 1
8e
t
#
:
x7d,x41xfx5dx55x1axdx1f,xfx38x37x51x51x29exp(At)x20x14x28x33x69x2f:
exp(At) = 16
"
2e3t +4e?3t 8e3t?8e?3t
e3t?e?3t 4e3t +2e?3t
#;
x5bx31x46x3fxfx51x5dx57x56x6x55x1ax20x29,x57x56x20x1cx31x10x6x55x1ax20x41x29x52
x53x49x31x5dx29,x64x53xfx51x5dx29c0(t),c1(t),x75x57x11x12x7cx60x1a,x53x2ax20x5d
xbx40x11x12x20x73x12x32x13x58x64x6cx29x20 exp(At) x20x33x69x2fxfx2fx14x67x10x52
x39.
x32,2)x4x73x122.3x121.4
exp(At) =
2
64 cos2t?
1
2 sin2t?
5
2 sin2t1
2 sin2t cos2t+
1
2 sin2t
3
75;
x3x5dx2x3det(DI?A)~y = ~f (t)x20x3fxd,x8x5d?D2 +4￠~y = ~f (t)x20
x3fxd~yp (t) =?14 (1;t)T,x7ax2f~xp (t) = (DI?A)?~yp (t) = 14 (?1?5t;2+t)T
x2fx3bx23x30x31x2x3x20x14x4ex3fxd,x75x57x2x3x20x3dxdx31
~x(t) = ~xp (t)+exp(At)~c:
x57x48x49xbx72x73x74x20xdx31
~x(t) = ~xp (t)+exp(At)(~x0?~xp(0))
x75x57x76x64x5dx20xdx31
~x(t) =
5
4 cos2t?
15
8 sin2t?
1
4?
5
4t;
1
2 cos2t+
7
8 sin2t+
1
2 +
1
4t
T
.
x32,3)x59x5dx29
exp(At) =
2
64 2e
t?1 1?et 1?et
3et?3 3?2et 3?3et
1?et et?1 2et?1
3
75;
x3x5dx2x3 det(DI?A)~y = ~f (t) x20x3fxd,x8x5d?D3?2D2 +D￠~y =
~f (t)x20x3fxd~yp (t) =
t; 12t2 +2t;t2 +4t
T
,x7ax2f
33
~xp (t) = (DI?A)?~y(t) =
3
2t
2 +2t?6; 9
2t
2 +5t?16;?3
2t
2?4t+3
T
x2fx3bx23x30x31x2x3x20x14x4ex3fxd.x75x57x2x3x20x3dxdx31~x(t) = ~xp (t)+exp(At)~c,
x57x48x49xbx72x73x74x20xdx31
~x(t) = ~xp (t)+exp(At)(~x0?~xp(0))
x76x64x5dx20xdx31~x(t) =
1+2t+ 32t2;5?5et +5t+ 92t2;5et?4?4t? 32t2
T
.
x32,4)x59x5dx29
exp(At) =
2
66
66
64
1
2e
t + 1
3e
2t + 1
6e
t et?e2t 2
3e
2t? 1
2e
t? 1
6e
t
1
2e
t? 1
2e
t et 1
2e
t? 1
2e
t
1
2e
t + 1
3e
2t? 5
6e
t et?e2t 2
3e
2t + 5
6e
t
3
77
77
75;
x4x7a ~f x2fx16x18x55x1ax460x59x2fx57x56Ax20x3fx5dx40x6bx52x53x68x70x5dx2
x3(DI?A)~x = ~f x20x16x55x1ax3fxd,x8xdx37x18x2x3x1b?A~xp = ~f x76x76
~xp =?(1;1;1)T x75x57x2x3x20x3dxdx31~x(t) = ~xp+exp(At)~c,x57x48x49xbx72
x73x74x20xdx31
~x(t) = ~xp +exp(At)(~x0?~xp)
x76x64x5dx20xdx31
~x(t) =
2
66
66
64
23e2t + 52et + 16e?t?1
5
2e
t? 1
2e
t?1
23e2t + 52et? 56e?t?1
3
77
77
75:
x32,5)x4x73x122.3x121.9
exp(At) =
2
66
4
cost+2sint?sint 2sint
cost?et +2sint et?sint 2sint
cost?et?3sint
2
et?cost+sint
2 cost?sint
3
77
5;
x3x5d(D?1)(D2+1)~y = ~fx76x3fxd~yp (t) = (?1;?1;?(1+t))T x61~xp (t) =
(DI?A)?~yp (t) = (1+2t;1+2t;?t)T
x2fx3bx23x30x31x2x3x20x14x4ex3fxd,x75x57x2x3x20x3dxdx31
~x(t) = ~xp (t)+exp(At)(~x0?~xp(0))
x8
~x(t) =
2
64 1+2t?cost1+2t?cost+et
t+ 12(cost+sint+et)
3
75:
34
4,x59x7cx60x57x56exp(A),x5dx29x2ax2bx20det(exp(A)).
1) A =
2
64 1 0 3?1 2 0
0 1?1
3
75,2) A =
2
64 1 4 23 1?1
2 1?2
3
75.
xd,x5bx31exp(At)x2fx2x3~x0 = A~xx20xdx57x56,x71Wronskyx3cx2bx2f
W (t) = det(exp(At)),x4Liouvillex21xf
det(exp(At)) = exp(0)exp
Rt
0 trAds
·
= exp
Rt
0 trAds
·
,x64x53
1) det(exp(A)) = e2,2)det(exp(A)) = e0 = 1.
5,x5dx29x2x3x000?2x00?x0 +2x = 0x20x3dxd.
6,x5dx29x1ex1dx2x3xbx40x11x12x00?3x0+2x = 2e?t;x(0) = 2;x0(0) =?1
x20xd.
xd,x6x5ax35x20x30x31x2x3x20xdx14x38x4fx2fe?t,x71x58?x31x62x21x20x16x18,
x37x1bx2x3x76,e?t2?3?+2￠ = 0,xdx29? =?1; 2,x52x46x5ax35x20xdx2f
x19x1ax29x79x20,x5bx5fx30x31x2x3x20x3dxdx31c1et+c2e2t,x3x6x23x30x31x2x3x20
x14x4ex3fxdx38x4fx2fce?t,cx31x62x21x16x18,x37x1bx76ce?t (1+3+2) = 2e?t,
xdx29c = 13,x64x53x2x3x20x3dxdx31x(t) = 13e?t + c1et + c2e2t,x30x7fx4x73
x74 x(0) = 2;x0(0) =?1 x52x53x7x21x16x18 c1 = 4,c2 =?73,x64x53xdx31
x(t) = 13e?t +4et? 73e2t.
7,x77x6x = ’(t)x2fx1ex1dx16x17x18x19x1ax0x1x2x3xbx40x11x12
x00 +ax0 +bx = 0; x(0) = 0; x0(0) = 1
x20xdx6bx2x70x = Rt0 ’(t?s)f(s)dsx2fx1ex1dx23x30x31x2x3
x00 +ax0 +bx = f(t)
x20xdx6bx71x58f(t)x31x7fx5cx20x0x1x4ax18.
x70x4ex68x70x5dx5ex6fx70x6bx5e
8*,x5ex60exp(At),x71x58
A =
2
66
66
66
64
3?1 1 1 0 0
1 1?1?1 0 0
0 0 2 0 1 1
0 0 0 2?1?1
0 0 0 0 1 1
0 0 0 0 1 1
3
77
77
77
75;
xd,x3fx5dx2x3x31?(2)5 = 0,x4x7a2x2fx1cx31x43x71(5x43x71),x7bx7cx60,
B = A?2I x20x7x314,x5bx5fx57x56Ax20Jordanx7x35x56x79x7ax3fx5dx402x38
x77x4eJordanx64x5,x66x77x4ex64x5xfx52x42x1x3ax2f1x1dx274x1dx20,x4fx1x3a
x2f2x1dx273x1dx20,x1cx5bB2 x20x7x312,x15B x20x7x11x12x12x1d,x64x53x66
x77x4e Jordan x64x5xfx42x2fx14x4e 2 x1d,x14x4e 3 x1d,x64x53x30x4dx6dx79x2fx31
35
(2)3,x54x7fx58x39x1ax7bexp(At) = e2t exp((A?2I)t) = e2t exp(Bt)x54
x7cx60x4cx3x7a(x3bx5bx46x2a),x6
exp(Bt) = c0I +c1B +c2B2 +c3B3,(1)
(x71x58B,B2 x7fx41x5dx30x4dx6dx79x2fx1fx76x69,x66x2fx3x7ax20x3bx5bx52x14),x61
8>
>><
>>>:
c0?2c1 +4c2?8c3 = e?2t
c0 = 1
c1 = t
2c2 = t2
(x53x44x2x3x1bx4cx2dxdx29,x66x2fx3x7ax20x3bx5bx52x1e)xdx29ck,k = 0;1;2;3,
x37x1b(1)x76
exp(At) = e2t
2
66
66
66
66
4
1+t?t t2 +t t2 +t 0 0
t 1?t t2?t t2?t 0 0
0 0 1 0 t t
0 0 0 1?t?t
0 0 0 0 1+e?2t2 1?e?2t2
0 0 0 0 1?e?2t2 1+e?2t2
3
77
77
77
77
5
:
9*,x71x72x2.2x58x21xf8x20Liouvillex2ex2fx70x17x4ex5ax7ax53!x31x47x48x20
x47x48x17x18x19x1ax2x3x1bx0 = A(t)xx20x41x18?jx27x3fx5dx28x18‰j; j = 1;2;￠￠￠;n;x35
x1x3ax48x49
nY
j=1
j = exp(
Z !
0
trA(s)ds)
x27
nX
j=1
‰j = 1!
Z !
0
trA(s)ds (mod 2…i! );
x2cx39x7a16x6fx53x6fx70.
x70x37x6Φ(t)x2fx55x14x7xdx56x6bx61Φ(t+!) = Φ(t)C,x6bt = 0,x61x19x14
x2fx52x4Liouvillex2ex2fx77x76x4e
nY
j=1
j = det(C) = det(Φ(!))det(Φ(0)) = exp(
Z !
0
trA(s)ds)
x5ax44x2fx77x78x6bx5ax18x3x49x53!x76x19x1ex2f.
x3c10*,x2x70x4ex5ax7ax53!x31x47x48x20x47x48x17x18x19x1ax2x3x1bx0 = A(t)xx20
x55x14x1fx7xdx56Φ(t)x6bx37x21x3x41x14x4ex3cx532!x31x47x48x20x38x57x56x4ax18P(t)x27
x14x4ex38x16x2x56Bx65x76Φ(t) = P(t)eBt.
36
x26 x27 2.4
1,x1cx1dx19x1ax23x30x31x2x3x20x3dxdx20x16x18x2ax2dx2ex2f.
x6nx1dx19x1ax23x30x31x2x3x(n) + a1(t)x(n?1) +￠￠￠+ an(t)x = f(t)x5ax35x20
x30x31x2x3x20x7x8xdx1bx31,x1(t),x2(t),￠￠￠,xn(t),x61x23x30x31x2x3x20x3d
xdx31
x(t) = xp(t)+c1x1(t)+c2x2(t)+￠￠￠+cnxn(t);
x71x58xp(t)x2fnx1dx19x1ax23x30x31x2x3x20x14x4ex3fxd,x52x39x2ax2bx29x1x33x34:
xp(t) =
Z t
t0
k(s;t)f(s)ds
W[x1(s);x2(s);￠￠￠;xn(s)]
x71x58k(s;t)x2fx2ax2bx4ax18x3cx2bx2f
flfl
flfl
flfl
flfl
flfl
fl
x1(s) x2(s) ￠￠￠ xn(s)
x01(s) x02(s) ￠￠￠ x0n(s)
￠￠￠ ￠￠￠ ￠￠￠ ￠￠￠
x(n?2)1 (s) x(n?2)2 (s) ￠￠￠ x(n?2)n (s)
x1(t) x2(t) ￠￠￠ xn(t)
flfl
flfl
flfl
flfl
flfl
fl
W[x1(t);x2(t);￠￠￠;xn(t)]x2fx1(t),x2(t),￠￠￠,xn(t)x20Wronskyx3cx2bx2f.
2,x5d nx1dx1fx16x17x18x19x1ax30x31x2x3L(D)[x] = 0x20x3dxd,x64x59xdx29
x3fx5dx2x3L(?) = 0x20x64x38x20x71?,x18?x2fkx43x1fx71,x61x30x31x2x3x38k
x4ex3fxdtse?t; s = 0;1;￠￠￠;k?1,x18? = fi§flix2fx14x5ax3ex0x20kx43x7ex71,
x61x30x31x2x3x382kx4ex3fxdtsefit cos(flt);tsefit sin(flt); s = 0;1;￠￠￠;k?1.
x14x3ex52x76x69nx4ex3fxd,nx1dx1fx16x17x18x30x31x19x1ax2x3x20x3dxdx0x2fx66n
x4ex3fxdx20x19x1ax1bx45.
3,x6nx1dx16x17x18x19x1ax0x1x2x3x31:
x(n) +a1(t)x(n?1) +￠￠￠+an?1x0 +anx = f(t),(1)
x5e(1)x20x3fx5dx6dx79x2fx31:
L(?) ·?n +a1?n?1 +￠￠￠+an?1?+an:
x3x5ex5dx5ex5fx60x70x2a:
D = ddt; D2 = d
2
dt2;￠￠￠;D
n = dn
dtn;
x61x2x3(1)x52x51x48x4fx2fL(D)[x] = f(t).
x5dx1fx16x17x18x23x30x31x2x3L(D)[x] = f(t)x20x3fxdx20x15x4cx17x18xe,x3d
x23x30x31x79x38x4fx2ff(t) = e?tpm(t),x57x71x58pm(t)x31tx20mx31x6dx79x2fx1f,
x52x53x39x15x4cx17x18xex5dx3fxd,x3d?x2fx5ax35x20x30x31x2x3x20kx43x3fx5dx40
x1f,x23x30x31x2x3x20x3fxdx14x38x4fx2fˉx(t) = tke?tqm(t),x57x3d?x59x2fx5ax35x20
x30x31x2x3x20x3fx5dx40x1f,x23x30x31x2x3x20x3fxdx14x38x4fx2f ˉx(t) = e?tqm(t),
37
qm(t)x2fx42x4emx31x6dx79x2f,x7ex71x37x1bx2x3x7f,x65x2x3x77x78x1exex79x20
x17x18x73xcx52x36x21x3fxd.
x41x37x1bx71x3x58,x52x53x58x39Leibnizxex61x3x2bx5dx5ex5fx60,x6g(t)x2f
nx1dx52x0x4ax18,x61x38:
L(D)[e?tg(t)] = e?tL(D+?)[g(t)]:
4,x70x76x7fx5cx1ex1dx19x1ax30x31x2x3x00(t) + p(t)x0(t) + q(t)x(t) = 0x20
x14x4ex3fxdx1(t) 6= 0,x61x57x14x4ex6x52x19x1ax29x79x20x3fxdx2(t)x31
x2(t) = x1(t)
Z t 1
x21(s)e
Rs
p(u)duds:
x71x58x59x16x29x1x2ax2ax20x29x1xfxdx31xax29x4ax18x20x55x14x3bx4ax18.
5,Eulerx2x3,x64x3cEulerx2x3x0x2fx52x51x48x70x2ax20nx1dx0x1x2x3
( sx2fx5ax2ax1a,xx2fx5bx5cx4ax18):
snx(n) +a1sn?1x(n?1) +￠￠￠+an?1sx0 +anx = f(s):
x42x5ax2ax1ax20x2ax50s = §et,x8t = ln(currency1s),x7ax2f dtds = 1s,x75x57
dx
ds =
1
s
dx
dt =
1
sDx;
d2x
ds2 =
1
s2
d2x
dt2?
dx
dt
!
= 1s2D(D?1)x;
x39x18x18x19x1axex52x76
dmx
dsm =
1
smD(D?1)(D?2)￠￠￠(D?m+1)x:
x7ax2f,Eulerx2x3x52x2bx31x16x17x18x19x1ax2x3x54x5dxd,x71x3fx5dx2x3x31:
(1)￠￠￠(n+1)+a1?(1)￠￠￠(n+2)+￠￠￠+an?1?+an = 0:
1,x2dx3ax2ax2bx1bx1bx4ax18x2fx19x1ax73x79x3ex2fx19x1ax29x79x35
1) t2 +2t;3t2?1;t+4; 2) 1;sin2 t;cos2t;
3) t;0;et 4) sint;cost;sin2t; 5) t2;tjtj.
x32,1)x19x1ax29x79,2)x19x1ax73x79,3)x19x1ax73x79,4)x19x1ax29x79,5)x19x1a
x29x79
2,x5dx29x2ax2bx16x17x18x19x1ax2x3x20x3dxd:
1) x(4)?5x00 +4x = 0, 2) x(5)?4x(3) = 0,
3) x00 +2x0 +10x = 0, 4) x(3)?4x00 +5x0?2x = 2t+3,
5) x(4)?2x00 +x = t2?3, 6) x00 +x0?2x = 8sin2t,
7) x00 +2ax0 +a2x = et, 8) x00?2x0 +3x = e?t cost,
9) x00 +x = sint?cos2t, 10) x00?4x0 +4x = et +e2t +1,
11) x00?2x0 +2x = tet cost, 12) x00 +x = csc3 t:
38
x32:
1)x3fx5dx2x3x31?42 +4 = 0,x71x5dx9x3fx5dx40x31? =?2;?1; 1; 2.
x64x53x(t) = c1e?2t +c2e?t +c3et +c4e2t,
2) x(t) = c1e?2t +c2e2t +c3 +c4t+c5t2,
3) x(t) = e?t [c1 cos3t+c2 sin3t],
4)x3fx5dx2x3x31L(?) =?3?4?2+52 = 0,x71x64x38x20x3fx5dx40x2f? = 1(x1e
x43),? = 2,x23x30x31x79x20x4fx2fx2f f(t) = 2t + 3 = e0t(2t + 3),x5b 0 x59
x2fx3fx5dx2x3x20x71,x64x53x3fxdx14x38x4fx2f ˉx(t) = at + b,x37x1bx2x3x5dx29
a =?1;b =?4,x75x57x3dxdx31x(t) =?t?4+et [c1 +c2t]+c3e2t,
5) x(t) = t2 +1+e?t [c1 +c2t]+et [c3 +c4t],
6)x3fx5dx2x3x31 L(?) =?2 + 2 = 0,x71x64x38x20x3fx5dx40x2f? =?2,
= 1,x23x30x31x79x20x4fx2fx2ff(t) = 8sin(2t) = Im8e2it,x5b2ix59x2fx3fx5dx2
x3x20x71,x64x53x3fxdx14x38x4fx2fxp(t) = Im?ce2it￠,x71x58cx2fx62x21x20xdx16
x18,x5bx5f,x59x5dx2x3L(D)[z(t)] = 8e2itx20xdx40x3fxd,x71x58z(t) = ce2it,x4
Leibnizxex61,
L(D)[z(t)] =
e2it
·
L(D+2i)[c] =
e2it
·
(2i?6)c = 8e2it;
x5dx29 c =?i+35,x75x57x3fxdx31 xp(t) = Imz(t) = Im
i+35 e2it
=
25 [cos2t+3sin2t],x64x53x3dxdx31 x(t) =?25 [cos2t+3sin2t] + c1e?2t +
c2et,
7) x(t) = 1(a+1)2et +e?at [c1 +c2t]; a 6=?1,
x(t) =
1
2t
2 +c1t+c2
et; a =?1,
8) x(t) = 141e?t [5cost?4sint]+et
h
c1 cos
p
2t
·
+c2 sin
p
2t
·i
,
9) x(t) = 13 cost+cost
c1? 12t
+c2 sint,
10) x(t) = 14 +et +e2t
c1 +c2t+ 12t2
,
11) x(t) = 14tet [cost+tsint]+et [c1 cost+c2 sint],
12) (x7d,x8x12x59x42x39x15x4cx17x18xex57xfx42x39x16x18x2ax2dx2ex2fx5dx3fxd)
x(t) = 12 csct+c1 cost+c2 sint.
3,x5dxdx2ax2bEulerx2x3x20x3dxd( sx2fx5ax2ax1a,xx2fsx20x5bx5cx4ax18)
1):s2x00 +sx0?x = 0, 2):s2x000?2x0 = 0,
3):s2x00?4sx0 +6x = s, 4):s2x00?sx0 +2x = slns,
x32,1),x(s) = c1s +c2s,
x32,2),x(s) = c1 +c2 ln(§s)+c3s3,
x32,3),x(s) = 12s+c1s2 +c3s3,
39
x32,4),x(s) = slns+s[c1 cos(lns)+c2 sin(lns)].
4,x70x76x7fx5cx1ex1dx19x1ax30x31x2x3x00(t) + p(t)x0(t) + q(t)x(t) = 0x20
x14x4ex3fxd’(t) 6= 0,x2x58x39Liouvillex21xfx5dx29x2x3x20x3dxd.
xd,x55y = x0(t),x61x3bx2x3xcx1bx7ax19x1ax2x3x1b
x0 = y; y0 =?q(t)x? p(t)y,x6x2x3x1bx20x55x14xdx31(x(t);y(t))T =
(x(t);x0(t))T,x61W(t) = W[’(t);x(t)] = ’(t)x0(t)?’0(t)x(t)x31xdx57x56x20
Wronskyx3cx2bx2fx4Liouvillex21xf,W(t) = c2e
R
trA(t)dt
x8’(t)x0(t)?’0(t)x(t) = c2e?
R
p(t)dt,x36x2fx79x7ax(t)x20x14x1dx19x1ax2
x3,x35x39x16x18x2ax2dx2ex2f,x6bx5ax35x20x30x31x2x3x20x3fxdx31h(t) = ’(t),x76
x3dxd
x(t) = ’(t)
c1 +c2
Z t 1
’2(s)e
Rs
p(u)duds
5,x7fx5cx2ax2bx1ex1dx19x1ax2x3x64x5ax35x20x30x31x2x3x38x14x4etx20x6dx79
x2fx3fxd,x2x5dx29x71x3dxd:
1)x00t2 lnt?tx0 +x = t2 ln2 t, 2)(2t+1)x00 +4tx0?4x = 4t2 +4t+1,
3)
t2 +t
·
x00+(t+2)x0?x = t+1t, 4)(2t+1)x00+(2t?1)x0?2x = t2+t:
x32,1) x(t) = c1t+c2 (1+lnt)+ t
2
2 (ln(t)?1),(x30x31x3fxdt)
x32,2) x(t) = c1t+c2e?2t +t2 + 14,(x30x31x3fxdt,x23x30x31x3fxdt2 +1=4)
x32,3) x(t) = c1 (t+2)+c2t?1 + 12 [3+(t+2)ln(§t)],(x30x31x3fxdt+2)
x32,4) x(t) = c1 (2t?1) + c2e?t + 12
t2 +1
·
.(x30x31x3fxd2t?1,x23x30x31
x3fxd(t2 +1)=2 )
6,x39x1cx3dx20x1dx1dxex5dxdx2ax2bx1ex1dx2x3(x5ax2ax1ax2ft):
1) x00 = 12x0, 2) xx00?(x0)2 +(x0)3 = 0,
3) x00 + 21?x?x0￠2 = 0, 4) x00 +
q
1?(x0)2 = 0,
5) ax00 +
h
1+(x0)2
i3=2
= 0; a 6= 0, 6) x00? x
0
t +
x0￠2 = 0:
x32,1)x = §23
q
(t+c1)3 +c2,
x32,2)t = x+c1 ln(c2x),x3ex38xdx = c,
x32,3)x = 1+ 1c
1t+c2
,
x32,4)x49x1x17x63x20x77x1exdx = c§tx22x3ex38x6x66x77x1exdx73x1fx70x57x48x20
40
x21x45x39x43x31Rx20xax2cxdx1e:
x =
8>
<
>:
c2 +(t?c1 +…=2); t? c1?…=2;
c2 +cos(t?c1); c1?…=2 < t < c1 +…=2;
c2?(t?c1?…=2); t? c1 +…=2:
x32,5)x = c2 +a
s
1?
t?c
1
a
2
,
x32,6)x = ln(c1 +c2t2).
7,x6’(t)x31x2x3x00 + k2x = f (t)x20xd,x71x58kx31x16x18,x4ax18f (t)
x41Rx44x0x1,x70x17:
1)x3dk 6= 0x1fx3dxdx31
’(t) = c1 cos(kt)+c2sin(kt)k + 1k
Z t
0
sin(k(t?s))f (s)ds,
2)x3dk = 0x1fx3dxdx31
’(t) = c1 +c2t+
Z t
0
(t?s)f (s) ds:
x70,x68x70x5dx5ex6fx70(x5e),x67x46x8x73x12x1912x12xdx32.
8.x6’1 (t);’2 (t)x31x00+p(t)x0+q(t)x = 0x20x77x4ex19x1ax29x79xd,x70
x17x2x3x00 +p(t)x0 +q(t)x = f (t)x20x55x14xdxdx52x33x34x31
x = c1’1 (t)+c2’2 (t)+
Z t
t0
’1 (s)’2 (t)?’1 (t)’2 (s)
W (s) f (s) ds
x71x58W (t) = W [’1 (t);’2 (t)]x31Wronskyx3cx2bx2f,c1;c2 x31x55x56x16x18.
x70,x68x70x5dx5ex6fx70(x5e).
9,x5ax7ax1ex1dx19x1ax2x3
x00 +2nx0 +!2x = 0; (n > 0; ! > 0x2fx16x18)
x2x0n < !;n = !x27n > ! xdx4ex12x4fx20x10x71x3dxdx3dt ! +1x1fx20x1a
x7b.
x32,x3fx5dx2x3x31?2 +2n?+!2 = 0,x3dx17x18n > 0,! > 0x1fx3fx5dx40
x20x1fx9x24x4dx7ax24,x64x53x55x14x3dxdx3dt ! +1x1fx24x23x24x24.
10,x5ax7ax1ex1dx19x1ax2x3
x00 +2nx0 +!2x = hsin(pt); (! > 0;! > n? 0; h >x2fx16x18)
x2x20x10x66x4ex2x3x20x3dxdx41x2ax2bx12x13x2ax3dt ! +1x1fx20x1ax7b:
1) n = 0x46p 6= !x4fn = 0x46p = !x20x12x4f:
2) n > 0x46p 6= !x4fn > 0x46p = !x20x12x4f.
41
xd,1)n = 0x46p 6= !x1f,x3dxdx31x(t) = c1 sin(!t+c2)+ h!2?p2 sin(pt),
xdx2fx77x4ex36x21x22x20x6ex6f,xcx23x1fx43tx20x2bx6f,x22x24x59x25x11.
x3dn = 0x46p = !x20x12x4f,x3dxdx31x(t) = c1 sin(!t+c2)? h2!tcos(!t),
xcx23x1fx43t ! +1,x22x24x29x2ax48x2bx2c,x28x27x64x3cx20x3ex26x16x7e.
2) n > 0x46p 6= !x1f,x3dxdx31
x(t) = c1e?nt sin(flt+c2)+ hH sin(pt? ),x71x58,fl = p!2?n2,
H =
q
(!2?p2)2 +4n2p2,x2fx16x18,cos = !
2?p2
H,sin =
2np
H xdx2fx14x4ex25x11x20x36x21x22x27x14x4ex36x21x22x20x6ex6f
,xcx23x1fx43t ! +1,xd
x27x37x48x23x24x66x4ex36x21x22.
n > 0x46p = !x20x12x4f.x3dxdx31
x(t) = c1e?nt sin(flt+c2)+ h2n! sin
!t? …2
,
xdx2fx14x4ex25x11x20x36x21x22x27x14x4ex36x21x22x20x6ex6f,xcx23x1fx43t ! +1,
xdx23x24x66x4ex36x21x22.
11,x5dxdx7ax28x5fx61x2x3 `00 + !2 sin` = 0,(x71x58 !2 = g=l x2fx36x16
x18)x20x4dx26x24x37x38x2x3x20xbx40x11x12:
`00 +!2` = 0; `(0) = `0 6= 0; `0(0) = 0
x32,`(t) = `0 cos(!t)
12,x77x6x = ’(t)x2fx1ex1dx16x17x18x19x1ax0x1x2x3xbx40x11x12
x00 +ax0 +bx = 0;x(0) = 0;x0(0) = 1
x20xd,x2x70x(t) = Rt0 ’(t?s)f (s)dsx2fx1ex1dx23x30x31x2x3
x00 +ax0 +bx = f (t)
x20x14x4ex3fxd,x71x58f (t)x31x7fx5cx20x0x1x4ax18.
x70,x0(t) = ’(0)f (t)+Rt0 ’0(t?s)f (s) ds = Rt0 ’0(t?s)f (s)ds,
x00(t) = ’0(0)f (t)+Rt0 ’00(t?s)f (s) ds = f (t)+Rt0 ’00(t?s)f (s) ds,x64
x53
x00(t)+ax0(t)+bx(t) =
= f (t)+Rt0 [’00(t?s)+a’0(t?s)+b’(t?s)]f (s) ds = f (t).x70x1d.
12*,x6fx70x29x1-x0x1x2x3
x(t) = w(t)+
Z b
a
G12(t;s)f(s;x(s);x0(s))ds, (1)
x41xdx29x73x1ex56x45x2ax6x78x40x11x12
x00 = f(t;x;x0); x(a) = fi; x(b) = fl, (2)
42
x20xcx1bx1a,x53x67x6fx70x2ax7cx2f
Z b
a
jG12(t;s)jds? (b?a)
2
8 ;
Z b
a
flfl
flfl@G12
@t (t;s)
flfl
flflds? b?a
2, (3)
x20x36x36x1a.
xd,x5e.
13*,x5dx29S-Tx11x12
(
u00 +?u = 0; t 2 (0;1),
u(0) = u0(1) = 0,
x20x3fx5dx40x4bx2bx67x71x5ax35x20x3fx5dx4ax18x4bx2b.
xd,1)? > 0 x1f,x2x3x20x3dxdx31 u = Asin(p?t + B),x4x78x40x73x74
u(0) = 0x76B = 0,x3x4x4x78x40x73x74u0(0) = 0,x76Ap?cos(p?) = 0,x8
x76x3fx5dx40x4bx2b?k = …2(12 + k)2,k = 0;1;2;￠￠￠:,x67x71x5ax35x20x3fx5dx4ax18
x4bx2b,uk = sin(p?kt).
2) 0x1f,x78x40x11x12x29xd(x2bx2cx6dx5ax70).
14*x5dxdx78x40x11x12
8
><
>:
x00?1 = 0; t 2 (1;2),
2x(1)+4x0(1) = 7;
x(2)+x0(2) = 5:
xd,x29x1x1ex31x76x2x3x20x3dxdx = 12t2 +c1t+c2,x4x78x40x73x74x76c1,
c2 x20x37x18x2x3,1+2c1 +2c2 +4(1+c1) = 7,2+2c1 +c2 +(2+c1) = 5,
xdx76,c2 = 1?3c1,x5bx5fx78x40x11x12x20xdx31,x = 12t2 +c1(t?3)+1,x36x38
x14x4ex55x56x16x18c1.
15*x5dxdx78x40x11x12
x00 = t, t 2 (0;1), x(0) = x(1) = 1:
xd,x29x1x1ex31x76x2x3x20x3dxdx = 16t3 +c1t+c2,x4x78x40x73x74x76c1,
c2 x20x37x18x2x3,c2 = 1,c1 + c2 = 56,xdx76,c1 =?16,x5bx5fx78x40x11x12x20
xdx31,x = 16(t3?t)+1,
43
x26 x27 3.1
1.x5ax7ax0x1x2x3 d~xdt = ~f(~x),x18x38 ~f(~x0) = 0,x61x44~x0 x31x2x3x20x2d
x2ex2f,x4fx30x2f,x5ax35x7axbx25x6ax20~x = ~x0 x2fx2x3x20xd,x44x31x2x3x20x31
x32x33,x3fx3ax48,x3d~x0 = 0x1fx44x31x2x3x20x34x33.
2.x18 ~f(~x) x0x1x52x0,x5e A · D~f(~x0) x31x4ax18x55x1a ~f(~x) x41xbx25x6a
~x0 x20x5ex60x64,(x4ax18x55x1ax20x5ex60x64x1cx44x4ax18x55x1ax20 Jacobi x56),x18
Ax20x3fx5dx40x20x1fx9x24x59x2fx24,x61xbx25x6ax20x25x21x1ax6x19x1ax37x38x2x3
d~x
dt = A(~x?~x0)x20xbx25x6a~x0 x20x25x21x1ax73x1e.
3,x1ex31x1fx17x18x6dx79x2f?2 +a1?+a2x20x71x24x14x38x35x1fx9x20x15x51x73
x74x2fa1 > 0,a2 > 0.
4,xdx31x1fx17x18x6dx79x2f?3 +a1?2 +a2?+a3 x20x71x24x14x38x35x1fx9x20
x15x51x73x74x2fa1 > 0,a2 > 0,a3 > 0,a1a2 > a3.
1,x5ax7ax23x21x16x17x1fx0 = x;y0 = y + t,x36x6fxbx37x59x2ax1ax27x38x1ax7b
x2fx30x48x62?
x32,x59x48x62.x5bx31xdx31x = c1et,y = c2et?(1+t).
2,x10x11xbx4x21x16x17x1f
x0 = y +x
1?x2?y2
·;y0 =?x+y
1?x2?y2
·;
x58x39xax6x7x2ax50,x = rcos ;y = rsin,x5dx29x36x20x6ex6ax3ex39x3ax27x7cx4e
x3ax3bx1ex20x2x3,x2cx3cx29x36x3fx20x1x3dx3ex7d.
xd,drdt = r
1?r2
·
,d dt =?1,x6ex6ax31x = 0,y = 0,x39x3ax31r = 1,x49
x6ex6ax22,x3ax3bx1ex2x3x20x67x18x2x3x31
r = 1p1+c
1e?t; =?t+c2:
x52x46x66x6dx3ax3bxcx23x1fx43x20x2bx6f,x24x2fx3fx1fx40x2x24x41x42x3bx6ax43x44,x49
x1x39x3a r = 1x22,x24x23x24x39x3a r = 1,x2ex45x67x18x76x49x6ex6ax22x20x3ax3bx1e
x2x3
r = 1p1+ce ;
x7dx5e.
3,x2x70x53x2ax10x2e 1)x18lx2fx5ax46x17x1fx20x39x3a,x61?l = Al = l.
2)x18x38x4cx3ax3blx20!(fi)xax2ax29x45?(A)x2fxfx4x14x6aQx75x48x20,
x8?(Al) = fQg,x61 Q x37x31x6ex6a,x2cx46x3d t ! +1(?1) x1fx3ax3b l
x23x24x7ax66x4ex6ex6a,x7cx52,x18x3dt ! +1(?1)x1fx3ax3blx23x24x7ax14x4ex6a
Q 2 G,x61Qx37x31x6ex6a,x46x38?l (A) = fQg.
x70,1x2x6x39x3alx20x47x48x31T,x55x6bx39x3alx44x14x6a~x(t),x61~x(t+kT) =
~x(t),k 2 Z,x64x53x39x3a l x44x55x14x6ax2fx39x3a l x20 !(fi) xax2ax6a,x5bx5f
l ‰?l (Al),x57x14x2x4,x4x7ax39x3a l x2fx38x4cx39x29,l x44x20x55x56x29x53x6a
x2b~x(tk)x20xax2ax6ax24x49x7al,x8?l (Al) ‰ l,x64x53?l = Al = l.
44
2)x5bx31?(Al) = fQg,x57fQgx2fx4x18x4ax7cx73x3ax3bx1bx48x20x29x45,x5b
x5fQx2fx6ex6a,x46x4x7axax2ax29xfx38x14x6a,x64x53x3dt ! +1(?1)x1fx3ax3b
lx23x24x7ax66x4ex6ex6a,x7cx52,x18x3dt ! +1(?1)x1fx3ax3blx23x24x7ax14x4ex6a
Q 2 G,x61lx20xax2ax29xfx4x14x6aQ 2 Gx75x48.x4x6cx70x36x37x31x6ex6a.
4,x2x70,x55x3bx39x3a lx24x3x41x14x4ex63x41x4dx39x3ax20x4bx39,x65x76x41x5f
x4bx39x3ax59x3x41x6ex6a.
x70,x5bx31x41x38x4cx29x45l x44x17x14x6a~x,x24x38~x0 = ~f (~x) 6= 0,x4 ~f (~x)
x20x0x1x1a,x3x41x63x41lx20x15x1x4dx20x4bx39,x65x41x5fx4bx39x58 ~f (~x) 6= 0,x8
x41x5fx4bx39x58x59x3x41x6ex6a.
5,x69x29x2ax2bx2x3x1bx20x6ex6a,x2dx3ax41x6ex6ax4cx20x14x31x37x38x2x3x1bx20
x6exdx20x25x21x1a:
1) x0 = x(1?x?y); y0 = y(2?3x?y),
xd,x6ex6ax31(0;0),(0;2),(1;0),(1=2;1=2),x55x1ax4d ~f (x;y)x20Jacobix56
x31A =
"
1?2x?y?x
3y 2?3x?2y
#
,x6ex6a(0;0)x4c,A =
"
1 0
0 2
#
,x3fx5d
x40x2cx7ax24,x6ex6ax59x25x21.
x6ex6a(0;2)x4c,A =
"
1 0
6?2
#
,x3fx5dx40x24x4dx7ax24,x6ex6ax27x37x25x21.
x6ex6a(1;0)x4c,A =
"
1?1
0?1
#
,x3fx5dx40x24x4dx7ax24,x6ex6ax27x37x25x21.
x6ex6a(1=2;1=2)x4c,A = 12
"
1?1
3?1
#
,2Ax20x3fx5dx2x3x31?2 +2
2 = 0,x38x2cx7ax24x20x3fx5dx40,x6ex6ax59x25x21.
2) x0 = 9x?6y +4xy?5x2;y0 = 6x?6y?5xy +4y2,
xd,x5dx6ex6a,x49x1x17x63x20x6ex6a(0;0)x22,x4
3(3x?2y) = x(5x?4y),6(x?y) = y(5x?4y),
x2ex455x?4yx76(y?2x)(x?2y) = 0,x8y = 2xx4fx = 2y,x75x57x1x3a
x76x6ex6a(1;2)x27(2;1).
x55x1ax4dx20Jacobix56x31A =
"
9+4y?10x 4x?6
6?5y 8y?5x?6
#
,x41x6ex6a(0;0)x4c
A = 3
"
3?2
2?2
#
,A3 x20x3fx5dx2x3?22 = (2)(?+1) = 0x38x2c
x7ax24x20x3fx5dx40,x6ex6ax59x25x21.
x41x6ex6a(1;2)x4cA =
"
7?2
4 5
#
,x3fx5dx2x3?2?12?+27 = (3)(9) =
0x38x2cx7ax24x20x3fx5dx40,x6ex6ax59x25x21.
x41x6ex6a(2;1)x4cA =
"
7 2
1?8
#
,x3fx5dx2x3?2+15?+54 = (?+6)(?+9) =
0,x3fx5dx40x24x4dx7ax24,x6ex6ax27x37x25x21.
45
3) x0 = y?x; y0 = y?x2?(y?x)
y2?2xy + 23x3
·
.
xd,x6ex6ax31(0;0)x67(1;1).
x41x6ex6a(0;0)x4cx55x1ax4dx20Jacobix56x31A = 3
"
1 1
0 1
#
x38x36x20x3f
x5dx401,x6ex6ax59x25x21.
x41x6ex6a(1;1)x4cA =
"
1 1
53 23
#
,x3fx5dx2x3?2 + 13? + 1 = 0x20x3fx5dx40
x20x1fx9x24x2fx35x20,x56x6ex6ax27x37x25x21.
6,x2dx3ax2ax2bx19x1ax0x1x2x3x1b~x0 = A~xx24xdx20x25x21x1a:
1) A =
2
64 0 1 11 0 1
2 2 1
3
75; 2) A =
2
64 1 2?22 1?2
3 2?3
3
75; 3) A =
2
64?3 2 2?3?1 1
1 2 0
3
75;
4) A =
2
64 2 0?11?1 0
3?1?1
3
75; 5) A =
2
64 0?1?1?1?1 0
1 0?1
3
75; 6) A =
2
64 0 1?11 0?1
2 2?3
3
75.
x32,1)x3fx5dx2x3x31?32?5?+5 = 0,x36x38x36x20x3fx5dx40(? = 1),
x64x53x24xdx59x25x21.
x32,2) x3fx5dx2x3x31 (?+1)2 +1￠ = 0,x3fx5dx40x20x1fx9x24x59x2cx7a
x24,x2cx46x3fx5dx40x20x1fx9xcx7ax24x20x3fx5dx40x24x2fx14x43x20,x64x53x24xdx25x21.
x32,3)x3fx5dx2x3x31?3 + 4?2 + 9? + 10 = 0,x36x20x17x18x24x2fx36x20,
x57x46x17x18x48x49x79x17 a1a2 = 4 ￡ 9 = 36 > a3 = 10,x64x53x3fx5dx40x20
x1fx9x24x4dx7ax24,x64x53x24xdx27x37x25x21(x1fx20x44x3fx5dx6dx79x2fx52x1xdx31
(?+2)2 +2?+5￠.)
x32,4)x57x56Ax20x1dn = 3,x3fx5dx2x3x31?3 = 0,? = 0x31xdx43x3fx5d
x40,x4bx3fx5dx57x56?I?A = Ax20x7= 2 6= n?3 = 0,x64x53x24xdx59x25x21.
x32,5)x3fx5dx2x3x31?(?+1)2 = 0,x3fx5dx40x24x59x2cx7ax24,x2cx46x3fx5d
x40? = 0x2fx14x43x20,x64x53x24xdx25x21.
x32,6)x3fx5dx2x3x31(?+1)3 = 0,x3fx5dx40x24x2fx35x20,x64x53x24xdx27x37
x25x21.
46
x26 x27 3.2
x5ax7ax1ex5fx16x1fx17x18x19x1ax2x3x1b~x0 = A~x+ ~f,x71x58Ax2fx23x6ex6bx1e
x1dx1fx2x56x6b ~f x2fx1ex5fx1fx16x55x1a,x61x2x3x38x34x14x20x6ex6a,x66x6ex6ax20xe
x5x31x4e
1)x3d A x20x77x3fx5dx40x23x24,x1ex6x1fx6ex6ax2fx75x6ax6bx3fx5dx40x24x31x36x1f
x2fx59x25x21x75x6ax6bx5ax35x20x21x16xdx2fx59x25x21x20; x3fx5dx40x24x31x35x1fx6ex6a
x31x25x21x75x6ax6bx5ax35x20x21x16xdx2fx27x37x25x21x20,(x75x6ax1cx52x4fx1x31x36x16
x75x6a(x3dx77x3fx5dx40x1ex6x4bx59x73xcx1f);x60x4fx75x6a(x3dAx2fx18x1ax57x56?I,
6= 0x1f); x50x2bx75x6a(x3dAx59x2fx18x1ax57x56?I,x4bx77x3fx5dx40x2fx73xcx20
x23x24x1fx18x1f).
2)x3dAx20x77x3fx5dx40x23x24,x6bx6x1fx6ex6ax2fx51x6ax6bx5ax35x20x21x16xdx2fx59
x25x21x20;
3)x3dAx20x77x3fx5dx40x2fx1fx9x23x24x20x14x5ax3ex0x7ex18x1f,x6ex6ax2fx52x6ax6b
x1fx9x31x36x1fx6ex6ax2fx59x25x21x52x6ax6bx5ax35x20x21x16xdx2fx59x25x21x20; x1fx9
x31x35x1fx6ex6ax31x25x21x52x6ax6bx5ax35x20x21x16xdx2fx27x37x25x21x20.
4)x3dA x20x77x3fx5dx40x2fx14x5ax23x24x20x3ex0x8x7ex18x1f,x6ex6ax2fx58x53x6b
x5ax35x20x21x16xdx25x21x4bx59x2fx27x37x25x21x20.
xbx4x68xcx6x7x17(x;y)x58x20x0x1x2x3x0 = f(x;y),y0 = g(x;y)x2b
x31xax6x7(r; )x58x20x0x1x2x3x20x2ex2f:
r0 = [xf(x;y)+yg(x;y)]r?1,0 = [xg(x;y)?yf(x;y)]r?2
x71x58x = rcos,y = rsin,
1,x5dx29x2ax2bx2x3x20xbx25x6a(x6ex6a),x2cx2dx3ax71xex5x27x25x21x1a:
1) x0 =?4x?y?2; y0 = 2x?y +4;
xd,x6ex6a(?1;2),x3fx5dx2x3?2+5?+6 = 0,x3fx5dx40? =?2;?3,x6e
x6ax31x25x21x75x6a.
2) x0 = 3x+4y?2; y0 = 2x+y?3,
xd,x6ex6a(2;?1),x3fx5dx2x3?2?45 = 0,x3fx5dx40? = 5;?1,x6e
x6ax31x51x6a(x59x25x21).
3) x0 = x?2 y0 = y?1.
xd,x6ex6a(2;1),x3fx5dx2x3?2?2?+1 = 0,x3fx5dx40? = 1,x73x35x20x3f
x5dx57x56x20x7x31x24,x56x6ex6ax31x59x25x21x60x4fx75x6a(x6ex75x6a).
4) x0 = 2y?3x?2; y0 = y?2x+1,
xd,x6ex6a(4;7),x3fx5dx2x3?2 + 2? + 1 = 0,x3fx5dx40? =?1,(x1ex43),
x73x35x20x3fx5dx57x56x20x7x31x14,x64x53x6ex6ax31x25x21x7ax24x75x6a(x50x2bx75x6a).
5) x0 = x?y?3; y0 = y?4x+1,
xd,x6ex6a(2;?1),x3fx5dx2x3?2?23 = 0,x3fx5dx40? =?1; 3,x6e
x6ax31x51x6ax53x59x25x21).
47
6) x0 = 2x?7y +19; y0 = x?2y +5
xd,x6ex6a(1;3),x3fx5dx2x3?2 + 3 = 0,x3fx5dx40? = §p3i,x6ex6ax31x58
x53x6a.
7) x0 =?x?y +1; y0 = x?y?5.
xd,x6ex6a(3;?2),x3fx5dx2x3?2 + 2?+ 2 = 0,x3fx5dx40? =?1§i,x6e
x6ax31x25x21x52x6a.
8) x0 = 4x?3y?1; y0 = 2x?y?1.
xd,x6ex6a(1;1),x3fx5dx2x3?2?3? + 2 = 0,x3fx5dx40? = 1; 2,x6ex6a
x31x59x25x21x75x6a.
2,x36x21x2ax2bx2x3x1bx20xax2ax4b,x2cx2dx3ax71x25x21x1a:
1)
8
<
:
x0 =?x+(x?y)
q
x2 +y2;
y0 =?y +(x+y)
q
x2 +y2:
x32,x41xax6x7x2a,x38x2x3 drdt = r?r2?1·,d dt = r x5bx5fx17x1fx38x39
x3a r = 1,x71x72x41r = 1x20x54x37 drdt x20x36x35x6x52x53x2dx21x66x39x3ax2fx59x25
x21xax2ax4b.
2)
8
>>>>
>>>>
<
>>>>
>>>
>:
x0 =
8
><
>:
y?x?x2 +y2?1￠p
x2 +y2 ; (x;y) 6= (0;0);
0, (x;y) = (0;0):
y0 =
8>
<
>:
x?y?x2 +y2?1￠p
x2 +y2 ; (x;y) 6= (0;0);
0, (x;y) = (0;0):
x32,x41xax6x7x2a,x38x2x3 drdt = 1?r2,d dt =?1r x5bx5fx17x1fx38x39x3a
r = 1,x71x72x41r = 1x20x54x37 drdt x20x36x35x6x52x53x2dx21x66x39x3ax2fx25x21xa
x2ax4b.
3)
(
x0 = x?x2 +y2?1￠?x2 +y2?9￠?y?x2 +y2?4￠,
y0 = y?x2 +y2?1￠?x2 +y2?9￠+x?x2 +y2?4￠:
x32,x41xax6x7x2a,x38x2x3 drdt = r
r2?1
·?
r2?9
·
,d dt = r2?4,x5b
x5fx17x1fx38x39x3a
r = 1,x71x72x41r = 1x20x54x37 drdt x20x36x35x6x52x53x2dx21x66x39x3ax2fx25x21
xax2ax4b.x1exfx52x76r = 3x2fx59x25x21xax2ax4b.
4)
(
x0 =?y +x?x2 +y2?1￠2 ;
y0 = x+y?x2 +y2?1￠2,
x32,x41xax6x7x2a,x38x2x3 drdt = r?r2?1·2,d dt = 1x5bx5fx17x1fx38x39
x3a r = 1,x71x72x41r = 1x20x54x37 drdt x20x36x35x6x52x53x2dx21x66x39x3ax2fx55x25
48
x21xax2ax4b.
3,x2dx3ax2ax2bx2x3x2fx30x3x41xax2ax4b:
1) x0 = x+y +x3=3?xy2; y0 =?x+y +x2y +2y3=3
x32,x59x3x41,x5bx31x2x3x20x2x24x4dx20x56x22= 2+2x2 +y2 > 0.
2) x0 =?2x+y?2xy2; y0 = y +x3?x2y.
x32,x59x3x41,x5bx31x2x3x20x2x24x4dx20x56x22=1+x2 +2y2￠ < 0.
3) x0 = y?x+x3; y0 =?x?y +y3.
x32,x3x41,x5bx31x17x1fxfx38x34x14x20x6ex6a (0;0),(x5bx31x6ex6ax35x48x49x2x3
y = x(1?x2),x = y(y2?1),x49x1x3bx6ax22x20x6ex6ax35x48x49
1 = (1?x2)(y2?1),x671 = (1?x2)2 +(y2?1)2,x8
(1?x2)2?(1?x2)(y2?1)+(y2?1)2 = 0,x75x57x761?x2 = 0,y2?1 = 0,x3f
x40x6bx56xfx38x3bx6ax53x22x20x6ex6a)x41xax6x7x2ax38dr=dt =?r[1?r2(cos4 +
sin4 )],x4x7a1=2? cos4 +sin4? 1,x64x53drdt x41x53x3bx6ax31x53x20x15x1x4d
x20x57(0 < r < 1)x44x2fx35x20,x41x53x3bx6ax31x53x20x15x1x2cx20x57(r > p2)x44
x2fx36x20,x4x4bx39x21xf,x17x1fx41x4bx391? r?p2x3ax3x41x59x25x21xax2ax4b.
4,x20x10x2x3x1bx0 = ax+by; y0 = cyx20x6ex6axex5x67x71x25x21x1a,x71
x58a,b,cx31x1fx16x18x46ac 6= 0.
x32,x3d a < 0,c < 0,a 6= c x1f,x6ex6ax31x25x21x75x6a,(x58x24x75x6a),x3d
a = c < 0,b 6= 0 x1f,x6ex6ax31x25x21x7ax24x75x6a(x50x2bx75x6a),x3da = c < 0,
b = 0x1f,x6ex6ax31x25x21x60x4fx75x6a(x6ex75x6a),x18x44x7fx59xcx2fx6ax6,x61x73x35
x20x75x6ax31x59x25x21.x3dac < 0x1f,x6ex6ax2fx51x6a(x59x25x21).
5,x41x2ax2bRLC x52x59x2x3x58,x6R;L;C > 0x25x31x16x18,x2x20x10x71
x6ex6ax20xex5x2cx2dx3ax71x25x21x1a.
d2Q
dt2 +
R
L
dQ
dt +
1
LCQ = 0:
xd,x79x7ax6ex6a
Q; dQdt
= (0;0)x20x3fx5dx2x3x31?2 + RL?+ 1LC = 0,
x3fx5dx40x20x1fx9x24x4dx7ax24,x5bx5fx6ex6ax2fx25x21x20,x57x46x3dR < 2
q
L=Cx1f
x2fx25x21x52x6a,
x3dR? 2
q
L=C x1fx2fx25x21x75x6a.
6,x2x70,x5ax7ax1ex5fx17x1fx6bx18x41x39x3a‘x20x42x14x22x5ax39U x58x59x54x6e
x6a,x61x41U x3ax20x55x14x39x3ax37x65‘x63x54x41x71x3ax39x52x58.
x70,x39x7cx70xe.x18x41U x3ax38x14x39x3axfx65‘x63x54x41x71x3ax39x52x58,x61
x4dx39x3axfx52x42x51x7aU x58,x7ax2fx41x4dx39x3ax3ax37x38x6ex6a,x6x11x12x20x77x6
x3fx40.
49
x26 x27 3.3
x5ax7ax0x1x2x3 d~xdt = ~f(x),~f(~x0) = 0x20xbx25x6a(x6ex6a),x5ax35x20x21x16
xd~x = ~x0x20x25x21x1ax70x76x59x42x39x19x1ax37x38x2x3x54x2dx2ex1fx5bx39x53x2ax77
x4ex21xfx54x2dx2e:
x25x21x1ax21xf,x6Gx2f ~f(~x)x20x21x45x39x58x63x54xbx25x6a~x0x20x5ax39,x18
x38G?~x0 x44x20x52x0x4ax18V(~x),x46x6V(~x)x41Gx20x39x63 ˉG (x8Gx6G
x20x78x4cx20x2cx29)x44x0x1,x46V(~x0) = 0,x61x18V(~x)x2fGx44x20x36x21x4ax18,
x41Gx44x79x7ax2x3 d~xdt = ~f(~x)x20x5dx5ex18 dVdt = rV(~x)￠ ~f(~x)? 0,x61x2x3
x20xbx25x6a~x0 x2fx25x21x20,x71x58
(1)x18 dVdt = rV(~x)￠ ~f(~x) · 0,x61x2x3x20xbx25x6ax2fx25x21x20x4bx59x2fx27
x37x25x21x20,
(2)x18x29x45 M = f~x 2 ˉGjrV(~x) ￠ ~f(~x) = 0g x58x59x54x2x3x20xbx25x6a
~x = ~x0 x53x22x20x7cx73x3ax3b,x61xbx25x6ax2fx27x37x25x21x20.
x7d,x3dV(x)x41Gx44x36x21,x46x5dx5ex18 dVdt = rV(~x)￠ ~f(~x)x41Gx44x35
x21x1f,x66x1fM = f0gx66V x4ax18x1fx0x48x49x27x37x25x21x1ax20x73x74.
x59x25x21x1ax21xf,x6Gx2f ~f(~x)x20x21x45x39x58x63x54xbx25x6a~x0 x20x5ax39,
x6Bx2fx29x45f~x 2 GjV(~x > 0gx20x14x4ex23x5cx20x0x3dx1x5d,x46~x0 2 ˉB,x46
x6V(~x)x41B x44x0x1x52x0,x41B x20x39x63 ˉB x44x0x1,x46V(~x0) = 0,x18
x41Bx44x79x7ax2x3 d~xdt = ~f(~x)x20x5dx5ex18 dVdt = rV(~x)￠ ~f(~x)? 0,x46x29x45
M = f~x 2 ˉBjrV(~x)￠ ~f(~x) = 0gx58x59x54x2x3x20xbx25x6a~x = ~x0 x53x22x20x7c
x73x3ax3b,x61xbx25x6ax2fx59x25x21x20.
x7d,x3dx41Gx44V(~x) > 0,x46x5dx5ex18x41Gx44x2fx36x21x1f,x66x1fM = f0g,
x66V x4ax18x0x48x49x59x25x21x1ax21xfx20x73x74.
1,x5dx29x2ax2bx2x3x1bx20x64x38xbx25x6a(x6ex6a),x2cx20x10x73x35x21x16xdx20
x25x21x1a:
1) x0 = ln(1+y +sinx); y0 = 2+ 3p3sinx?8;
xd,x6ex6ax31(k…;0),k 2Z,x41x6ex6ax4cx20Jacobix56x31
A =
"
(?1)k 1
1
4(?1)
k 0
#;
x3dkx2fx6ex18x1f,x3fx5dx40? =?12,x25x21,x3dkx2fx5ex18x1f,x77x3fx5dx40x6bx6,
x59x25x21.
2) x0 = y; y0 =?x+y?x2￠;? > 0;
xd,x6ex6a(0;0),1;0￠,x3fx5dx2x3x1x3ax31?2+1 = 0x27
21 = 0,x3fx5dx40x38x36x20x1fx9,x64x53x6ex6ax24x59x25x21.
3) x0 = y?x; y0 = y?x2?(x?y)(y2?2xy + 23x3).
xd,1,x6ex6ax31(0;0),x41x6ex6ax4cx20Jacobix56x31
A =
"
1 1
0 1
#;
50
x38x36x20x3fx5dx401,x64x53x6ex6ax59x25x21.
2,x6ex6ax31(1;1),x41x6ex6ax4cx20Jacobix56x31
A =
"
1 1
53 23
#;
x3fx5dx2x3x31?2 + 13?+1 = 0,x3fx5dx40x24x14x38x35x1fx9,x27x37x25x21.
2,x5fx60x2ax2bx19x1ax0x1x2x3x24xdx20x25x21x1a:
1) x000 +5x00 +6x0 +x = 0;
x32,x3fx5dx2x3x31?3 + 5?2 + 6? + 1 = 0,a1 > 0;a2 > 0;a1a2 = 5￠6 =
30 > a3 = 1 > 0,x71x72Hurwitzx21xf,x71x20x1fx9x24x4dx7ax24,x64x53x24xdx27
x37x25x21.
2) x0 =?x?y; y0 =?y?z; z0 =?z?x,(?x31x16x18):
x32,x3fx5dx2x3x31
()3 +1 = (?+1)2?(1+2?)?+?2 +?+1￠ = 0,
x3d? >?12 x1fx24xdx59x25x21,? =?12x1f,x24xdx25x21,x3d? <?12 x1f,x24xd
x27x37x25x21.
3) x0 =?x?y +z; y0 = x?2y +2z; z0 = x+2y +z.
x32,x3fx5dx2x3?3 + 2?2?59 = 0x41x39x43(2;3)x58x38x36x71,x64x53
x24xdx59x25x21.
3,x20x10Van der Polx2x3
x00 +?
x2?1
·
x0 +x = 0; (? > 0)
x24xdx20x25x21x1a.
xd,x5bx31x3d? > 0x1f,x5ax35x20x2x3x20x19x1ax37x38x2x3x00x0+x = 0
x20x3fx5dx2x3?2+1 = 0x38x36x20x3fx5dx40,x64x53x3bx2x3x20x24xdx59x25
x21.
4,x20x10x2ax2bx23x19x1ax0x1x2x3x1bx24xdx20x25x21x1a:
1) x0 = y; y0 = a?1?x2￠y?bx;(a? 0;b > 0)
x32,x24xda > 0x1fx59x25x21,a = 0x1fx25x21 (x6bV(x;y) = bx2 +y2),x6e
x6a(0;0)x41a2? 4bx1fx2fx59x25x21x75x6a,0 < a2 < 4bx1f,x2fx59x25x21x52x6a.
a = 0x1fx2fx58x53x6a.
2) x0 = y; y0 =?ay?bsinx;(a? 0;b > 0)
x32:x24xda > 0x1fx27x37x25x21,a = 0x1fx25x21(x6bV(x;y) = b(1?cosx)+
1
2y
2),x6ex6a(0;0)x41a2? 4bx1fx2fx25x21x75x6a,0 < a2 < 4bx1f,x2fx25x21x52
x6a,a = 0x1fx2fx58x53x6a.
5,x2dx3ax2ax2bx4ax18x20x21x6x1a:
1) V (x;y) = x2, 2) V (x;y) = x2?2xy2,
3) V (x;y) = x2?2xy2 +x4 +y4, 4) V (x;y) = x2 +2xy +y2 +x2y2,
5) V (x;y) = xcosx+ysiny:
x32,1)x16x36 2)x2ax6 3x2x36x21 4x2x36x21 5x2x2ax6
6,x2x39x4fx70V (x;y) = ax2 +y2,(a > 0)x20Liapunovx4ax18x36x21x2a
x2bx2x3x1bx24xdx20x25x21x1a:
51
1) x0 =?xy2; y0 =?x2y.
xd:x6bx36x21x4ax18V (x;y) = x2+y2,x5dx5ex18dVdt =?4x2y2? 0,x64x53x2
x3x20x24xdx2fx25x21x20,x4bx59x2fx27x37x25x21x20.(x66x75x3ax3bx2x3x2?y2 = c2,
x3dt ! +1x1fx23x24x21x16xd(x;y) = (§c;0),x3ax3bx2x3x2?y2 =?c2,x3d
t ! +1x1fx23x24x21x16xd(x;y) = (0;§c)x9x29).
2) x0 =?x+xy2; y0 =?2x2y?y3.
xd,x6bx36x21x4ax18V (x;y) = 2x2 +y2,x5dx5ex18 dVdt =?2?2x2 +y4·x2f
x35x21x20,x64x53x2x3x20x24xdx2fx27x37x25x21x20.
3) x0 =?x+2y3; y0 =?2xy2.
xd,x6bx36x21x4ax18V (x;y) = x2 +y2,x5dx5ex18 dVdt =?2x2? 0,x64x53x2
x3x20x24xdx2fx27x37x25x21x20,x5bx31 dVdt = 0x20x29x45f(x;y)jx = 0gx58x49x1
x6ex6ax53x22x59x63x54x71x61x7cx73x3ax3b.
4) x0 = x3?2y3; y0 = xy2 +x2y + 12y3.
xd,x6bx36x21x4ax18V (x;y) = x2=2+y2,x5dx5ex18 dVdt =
x2 +y2
·2x2fx36
x21x20,x64x53x2x3x20x24xdx2fx59x25x21x20.
7,x5fx60x2ax2bx23x19x1ax2x3x1bx24xdx20x25x21x1a:
1) x0 =?x?y +(x?y)?x2 +y2￠; y0 = x?y +(x+y)?x2 +y2￠.
xd,x4x7ax19x1ax37x38x2x3x20x3fx5dx40?1§ix24x14x38x35x1fx9,x64x53x3bx2
x3x20x24xdx2fx27x37x25x21x20.
2) x0 =?y2 +x2?x2 +y2￠; y0 =?x2?y2?x2?y2￠
xd,x41x39x39E = f(x;y)jx+y < 0gx44V (x;y) =?(x+y) > 0,x5dx5ex18
dV
dt =
x2 +y2
·
x4 +y4
·x41x3bx6ax20x15x1x4dx20x5ax39x44x2fx36x21x20
.x57
(0;0) 2 ˉE,x5bx5fx24xdx2fx59x25x21x20.
3) x0 =?xy6; y0 = x4y3,
xd,x6bx36x21x20 V x4ax18 V (x;y) = x4 + y4,x5dx5ex18 dVdt · 0,x5bx5f
x24xdx25x21x20x4bx59x2fx27x37x25x21x20,(x5bx31x75x3ax3bx2x3x0x2f V (x;y) =
x4 +y4 = c2 x52x46)
4) x0 = ax?xy2; y0 = 2x4y, (ax31x67x18):
xd,x6bx36x21x20 V x4ax18 V (x;y) = x4 + y2,x5dx5ex18 dVdt = 4ax4,x3d
a? 0x1fx24xdx25x21,x4bx59x2fx27x37x25x21x20.x5bx31x38x21x16xd(x;y) = (0;c).
x3da > 0x1fx24xdx2fx59x25x21x20.x5bx31x3bx2x3x20x19x1ax37x38x2x3x38x36x20x3f
x5dx40a.
5) x0 = ax?y2; y0 = 2x3y,(ax31x67x18)
xd,x6bx36x21x20V x4ax18V (x;y) = x4+y2,x5dx5ex18dVdt = 4ax4,x3da = 0
x1fx24xdx25x21x4bx59x2fx27x37x25x21x20,x5bx31x38x3ax3bV (x;y) = x4+y2 = c2,x3d
a > 0x1fx24xdx2fx59x25x21x20,x5bx31x2x3x20x19x1ax37x38x2x3x38x36x20x3fx5dx40
52
a,x3da < 0x1fx24xdx2fx27x37x25x21x20,x5bx31 dVdt = 0x20x29x45f(x;y)jx = 0g
x58x49x1x6ex6a(0;0)x53x22x59x63x54x71x61x7cx73x3ax3b.
8,x6cx21x2x3x1b
x0 = y?xf (x;y); y0 =?x?yf (x;y),
x71x58 f (x;y) x5ax71x2ax1ax0x1x52x0,x2x70,x41x3bx6ax20x45x53x5ax39x3a,x18
f > 0,x61x24xdx31x27x37x25x21,x18f < 0,x61x24xdx59x25x21.
x70,x6bx36x21x20Liapunovx4ax18V (x;y) = x2 +y2,
x5dx5ex18 dVdt =?2
x2 +y2
·
f (x;y),x5bx5fx41x3bx6ax20x45x53x5ax39x3a,x18f >
0,x5dx5ex18x2fx35x21x20,x61x24xdx31x27x37x25x21x20,x18f < 0,x5dx5ex18x2fx36x21
x20,x61x24xdx59x25x21.
9,x6cx21x1ex1dx55x19x1ax2x3
x00 +f (x) = 0;
x71x58f (0) = 0,x46x3dx 6= 0x1fx38xf (x) > 0 (x5a?k < x < k;k > 0)x2x7e
x36x2bx31x1ex5fx14x1dx2x3x1b(x55x = x;y = x0),x2cx39x4fx70
V (x;y) = 12y2 +
Z x
0
f (s) ds
x20Liapunovx4ax18x20x10x2x3x1bx24xdx20x25x21x1a.
xd,x7ex1ex1dx2x3x2bx31x1ex5fx2x3x1bx70x2a:
x0 = y; y0 =?f (x):
x4x12x6x73x74,Liapunovx4ax18x2fx36x21x20,x5dx5ex18 dVdt = 0,x5bx5fx24xdx2f
x25x21x20.
10,x14x4dx62x64x63x64x48x65x41x14x4ex55x66x31Rx20x57x44(x57x67x68x7axfxbx4(
x?y xbx4),z x8x6x57x20x14x68x66x43x45,x3dx66x4ex57x53x16xcx68x22! > 0x42
(x69x68x24x44x20) zx8x67x61x1f,x62x64x20x5fx61x2x3x31
d2
dt2?
g
R sin?!
2 cos sin = 0;
x66x5x7exax6x7x6ax21x41x57x44(x53x57x53x31xax6ax53Ozx31xax8),x31xaxc,g >
0x31x43x6bx6fx68x22x16x18,x2x7ex62x64x5fx61x2x3x2bx48x1ex5fx14x1dx2x3x1b,x69
x29x71x2dx2ex2fx53x8x6ex6ax2,x2cx20x10xbx25x6ax20x25x21x1a.
xd,x6x =,y = x0,x61x2x3x2bx31x2x3x1b:
x0 = y; y0 = sinx
g
R +!
2 cosx
:
x6ex6ax31A = (0;0),B = (…;0),x3dg < !2Rx1f,x3ex38x6ex6aC = (xc;0) =
(§arccos(? g!2R;0),x6ex6aAx2fx51x6ax53x59x25x21x2,x3dg < !2Rx1f,x6ex6aB
53
x2fx51x6a(x59x25x21),C x2fx19x1ax37x38x2x3x20x58x53.x3dg > !2Rx1f,B x2fx19
x1ax37x38x2x3x20x58x53.x3dg = !2Rx1f,x6ex6aBx2fx23x36x61x6ex6a(x3fx5dx40x5d
x31x24).
x5ax7ax58x53x1fx20B,C,x27x23x36x61x6ex6ax20Bx20x25x21x1a,x52x6cx73x129x20
x2xex75x76Liapunovx4ax18x54x2dx21,x4bx71x72x6dxfx56x45x52x5c,x36x3fx24x35x4d
x2fx25x21x20,x1fx20x44,x3dg? !2Rx1f,x5ax7aBx6a,x6bx36x21x20V x4ax18(x1fx20
x44x2fx3ax3bx2x3x20x29x1)
V(x;y) = y2 +4sin2 x?…2
g
R?!
2 cos2 x?…
2
x57x5dx5ex18 dVdt · 0,x5bx5f,B x6ax2fx25x21x20.x3dg < !2R x1f,x5ax7a C x6a,
x6bx36x21x20 V x4ax18 (x1fx20x44x2fx3ax3bx2x3x20x29x1)x2cx46x10x11x69 g=R =
!2 cosxc,x46x3fx38x4e
V(x;y) = y2+2(cosx?cosxc) gR+!2(cos2 x?cos2 xc) = y2+!2(cosx?cosxc)2
x57x5dx5ex18 dVdt · 0,x5bx5f,C x6ax2fx25x21x20.
11,x2x3x1b
x0 = y?x3; y0 =?2
x3 +y5
·
x42x30x4x19x1ax37x38x2x3x7x21x71x24xdx20x25x21x1ax35x2x6ex69Liapunovx4ax18
x54xdx7x66x4ex2x3x1bx24xdx20x25x21x1ax11x12,x1ex1fx2ax61x2x3x20x1cx31x79x65
x44x2x3x20x24xdx31x59x25x21x20.
xd,x5bx31x19x1ax37x38x2x3x20x17x18x57x56x20x3fx5dx40x2fx24,x64x53x59x42x4x19
x1ax37x38x2x3x7x21x71x24xdx20x25x21x1a,x6bx36x21x20Liapunovx4ax18V (x;y) =
x4 + y2,x61x5dx5ex18 dVdt =?4
x6 +y6
·x2fx35x21x20
,x5bx5fx24xdx2fx27x37x25
x21x20,x2ax61x2x3x20x1cx31x79x65x44x2x3x48x31
x0 = y +x3; y0 =?2
x3?y5
·
x61x5dx5ex18 dVdt = 4?x6 +y6·x2fx36x21x20,x5bx5fx24xdx2fx59x25x21x20.
12,x41x14x6fx70x71x3ex3bx44x27x3x23x72x38(x73x6d)x27x74x38(x75x6d),x53x(t)x27y(t)x1
x3ax5ex41x1fx23tx72x27x74x20x4ex38x18x1a,x4x7ax72x76x3e,x74x76x72,x52x53x77x6x36
x3fx20x8x2bx77x2cx1x3ax311xdxdt = A?By; 1y dydt =?D+Cx,(x66x5A;B;C;D
x25x31x36x18,x? 0,y? 0)x8x66x4ex27x26x17x1fx20x18x18x4x5x31Volterra–Lotka
x75x73x2x3,8
><
>:
dx
dt = Ax?Bxy;dy
dt =?Dy +Cxy:
54
x2x5dx29x66x4ex17x1fx20x1fx20xbx25x6a(x8x6ex6a),x20x10x71x25x21x1a,x2cx1ex17x71
x1fx20x56x45.
xd,x1fx20x44,x72x27x74x20x18x1ax24x59x2fx24,x64x53x1fx20x20xbx25x6ax31z =
(DC; AB),x41xbx25x6ax20Jacobix56x31
2
64 0?
BD
CCA
B 0
3
75:
x3fx5dx2x3x2f?2 +AD = 0,x56xbx25x6ax2fx19x1ax37x38x2x3x20x58x53(x59x42x7
x21x3bx2x3x20x25x21x1a),x5dx29x3bx2x3x20x3ax3bx29x1:
V(x;y) · 1A
Cx
D?1?ln
Cx
D
+ 1D
By
A?1?ln
By
A
= const:
x52x46x41x19x14x7ex2ax44V(x;y)? 0,x16x41xbx25x6aV = 0,x5dx5ex18 dVdt = 0,
x64x53xbx25x6ax25x21,(x7d,x5bx31xbx2dx70x17es? 1 + s,s >?1x1fx59xcx2fx77
x78x6bx5ax18x76s? ln(1+s),x55s = p?1,x5bx5fp?1? lnpx6bx75x57V(x;y)
x41x19x14x7ex2ax44x23x35x6bx16x41xbx25x6ax31x24)
13*,x20x10x2ax2bx2x3x1bx20x1x5dx11x12.
1:x0 = x?2y; y0 = y?x2 +?:
xdx4ex6ex6ax2f(x;y) = ([1§(1+8?)1=2]=4;[1§(1+8?)1=2?4?]=8),x41
x6ex6ax20x19x1ax37x38x2x3x20x3fx5dx2x3x20x71x2fx4e?1 = 1+[2§(1+8?)1=2]1=2,
2 = 1?[2§(1+8?)1=2]1=2,x5bx5fx5ax7ax6ex6a(x;y) = ([1+(1+8?)1=2]=4;[1+
(1+8?)1=2?4?]=8)x41? 2 [?1=8;1)x1fx3dx3ex31x51x6a,? <?1=8x1fx6bx66x6e
x6ax2ex66,x5ax7ax6ex6a(x;y) = ([1?(1+8?)1=2]=4;[1?(1+8?)1=2?4?]=8)x41
2 [?1=8;0)x1fx3dx3ex31x51x6a,? <?1=8x1fx6bx66x6ex6ax2ex66,x41? 2 [0;3=8]
x1fx6bx6ex6ax31x59x25x21x75x6a,x41? > 3=8x1fx6bx6ex6ax31x59x25x21x52x6a,x5bx5f
x1x5dx40x31? = 3=8 (x75x6ax2ax52x6a,x25x21x1ax59x2a)? = 0 (x51–x75x1x5d),x67
=?1=8 (x6ex6ax2ex66).
2:x0 = y; y0 = [(x+1)2+y][(x?1)2 +?+y].
xdx4ex6ex6ax2f(x;y) = (?1§?1=2;0)x67(x;y) = (1§()1=2;0)x41x6e
x6ax20x19x1ax37x38x2x3x20x3fx5dx2x3x1x3ax31x4e
2?2(2currency12?1=2 +?)?currency14?1=2(2currency12?1=2 +?) = 0x67
2?2(2§2()1=2)?currency14()1=2(2§2()1=2) = 0.
x6ex6a(x;y) = (?1§?1=2;0)x3d? = 0x1fx2fx59x25x21x75x6ax6bx3d? > 0x1f
x2ax31x14x4ex51x6ax27x14x4ex59x25x21x75x6ax6bx5bx5fx1x5dx40x2f? = 0 (x51–x75x1x5d
x40)x6ex6a(x;y) = (1§()1=2;0)x3d? = 0x1fx2fx59x25x21x75x6ax6bx3d? < 0
x1fx2ax31x14x4ex51x6ax27x14x4ex59x25x21x75x6ax6bx5bx5fx1x5dx40x2f? = 0 (x51–x75x1
x5dx40).
3:x0 =x2; y0 = y(2x)
xdx4ex6ex6ax2f (x;y;?) = (0;0;c); (x;y;?) = (;0;?); x67 (x;y;?) =
(0;c;0),x41x6ex6ax20x19x1ax37x38x2x3x20x3fx5dx2x3x20x71x1x3ax31? = §?;
55
= 2? §?; x67? = 0,x3d? = 0x1fx6bx3ax19x2f,y = cx2 x67x = 0;y = c,
x3bx6ax2fx25x21x50x2bx75x6ax6b? 6= 0x1fx6bx3bx6ax2ax31x51x6ax57x6ex6a(0;c(6= 0);0)
x2ex66,x4bx75x3bx6ax1x29x75x6a(x;y;?) = (;0;?(6= 0)),? > 0x1fx2fx59x25x21
x75x6ax6b? < 0x1fx2fx25x21x75x6a;x5bx5fx1x5dx40x31? = 0 (x3fx40x4cx1x5dx40).
4:x0 =x+y +x=(1+x2 +y2); y0 =?xy +y=(1+x2 +y2).
xdx4ex7ex2x3x39xax6x7(r; )x2bx31x4e r0 = r[1=(1+r2)]; 0 =?1,
x52x46x3bx2x3xfx38x14x4ex6ex6a(x;y) = (0;0),x3d? 2 (?1;1)x1fx6bx6ex6ax2f
x59x25x21x52x6ax6bx3d? > 1x1fx6bx6ex6ax2fx25x21x52x6a,x46x16x3d? 2 (0;1)x1fx6bx38
x14x4exax2ax4bx53x25x21x2 r = [(1)=?]1=2,x5bx5fx1x5dx40? = 0x67? = 1x24
x2fHopfx1x5dx40,(x7d, 0x67 1x1fx29xax2ax4bx20x75x10x52x53x75x5dxa
x6x7x2ax20x2x3x76x69x6bx4ax52x75x5dx29x2x24x4dx20x56x22( = 2(1=(1+r2)2))
x76x29).
56