~ ?yvD ?Dy
? ??
?±sZ?
V?¨ú¨Z?
~ ?yvD ?Dy
Ba ?±sZ?#  pE
?l
0),(),( =+ dyyxQdxyxP5
dyyxQdxyxPyxdu ),(),(),( +=
?μ ?±s?
T
è ?
,0=+ ydyxdx
),(
2
1
),(
22
yxyxu +=Q
?±sZ?
 ?Z?
,),( ydyxdxyxdu +=∴
[
^ ?±sZ?,
.
x
Q
y
P
=
?±sZ?
~ ?yvD ?Dy
3E
0),(),( =+ dyyxQdxyxP
n?¨ wLsD
^?í1,
x
Q
y
P
=
Q
Y31
∫∫
+=
y
y
x
x
dyyxQxdyxPyxu
00
),(),(),(
0
,),(),(
00
0
xdyxPdyyxQ
x
x
y
y
∫∫
+=;),( Cyxu =
o¨°¤X ?±s￥ZE,
?±sZ?
~ ?yvD ?Dy
.
0)3()3(
2323
￥Y3
pZ? =?+? dyyxydxxyx
3,6
x
Q
xy
y
P
=?=
^ ?±sZ?,
∫∫
+?=
yx
dyyxdxyxyxu
0
3
0
23
)3(),(
.
42
3
4
4
22
4
C
y
yx
x
=+?
eZ?￥Y31
,
42
3
4
4
22
4
y
yx
x
+?=
è
~ ?yvD ?Dy
.0
32
4
22
3
￥Y3 pZ? =
+ dy
y
xy
dx
y
x
3,
6
4
x
Q
y
x
y
P
=?=
^ ?±sZ?,
|P
×?F
)
32
(
1
4
2
32
dy
y
x
dx
y
x
dy
y
+
)()
1
(
3
2
y
x
d
y
d +?=
.
1
3
2
C
y
x
y
=+?
eZ?￥Y31
),
1
(
3
2
y
x
y
d +?=
è 2
~ ?yvD ?Dy
=asy0E
?l

0),( ≠yxμ
?? V±f

PZ?
0),(),(),(),( =μ+μ dyyxQyxdxyxPyx ?1 ?
±sZ?5?
),( yxμ
1Z?￥ sy0 
ù5, ? pZ?￥sy0?
~ ?yvD ?Dy

TE
,
)()(
x
Q
y
P
=
μμ
Q
x
Q
x
Q
y
P
y
P
+
=
+
μ
μ
μ
μ

H]" μ
x
Q
y
P
y
P
x
Q
=
μμ lnln
p3? ?^
+
y1,;,μ1
HoD? xa μ
,0=
y
μ
,
dx
d
x
μμ
=
~ ?yvD ?Dy;,μ1
HoD? yb μ
)(
1ln
x
Q
y
P
Qdx
d
=∴
μ
)(xf=
.)(
)(
∫
=∴
dxxf
exμ
,0=
x
μ
,
dy
d
y
μμ
=
)(
1ln
y
P
x
Q
Pdy
d
=∴
μ
)( yg=
.)(
)(
∫
=∴
dyyg
eyμ
~ ?yvD ?Dy
43E
Y43X±s¤?
),( yxμ
èn￥ ?±sVr
T
+
=+
2
22
yx
dydyxdx?
=
x
y
d
x
ydxxdy
2
=
+
x
y
d
yx
ydxxdy
arctan
22
()xyd
xy
ydxxdy
ln=
+
+=
+
+
)ln(
2
1
22
22
yxd
yx
ydyxdx
+
=
yx
yx
d
yx
ydxxdy
ln
2
1
22
~ ?yvD ?Dy
Vê¨￥sy0μ
.,,
1
,
1
,
1
,
1
2222222
?
x
y
y
x
yxyxxyx ++
.0)()3(
22
￥Y3
p±sZ?
=+++ dyxyxdxyxy
3
,
1
)(
1
xx
Q
y
P
Q
=
Q
∫
=∴
dx
x
ex
1
)(μ,x=
è 3
5eZ?1
,0)()3(
2322
=+++ dyyxxdxxyyx
~ ?yvD ?Dy
,0)()3(
2322
=+++ dyyxxdxxyyx
)(3
32
xdyydxxydyxydxx +++
))(
2
1
(
23
xyyxd +=
,0=
eZ?￥Y31
.)(
2
1
23
Cxyyx =+
(
TE )
VFE
~ ?yvD ?Dy
.0)1(2
22
￥Y3=+ dyyxdxyxx
3
|Z?P
×?F,μ
è 4 p±sZ?
,022
22
=+ dyyxdxyxxxdx
,0)()(
2222
=+ dyyxxdyxxd
,0)()(
222
=+ yxdyxxd
eZ?￥Y31,)(
3
2
2
3
22
Cyxx =?+
~ ?yvD ?Dy
.0)1(ln2
222
￥Y3=+++ dyyyxydxxy
3 |Z?P
×?F,μ
,01)ln2
222
=+++ dyyydyxydxxy
,
1
),(
y
yx =μ^?
,01)ln2(
2
2
=+++ dyyydy
y
x
ydxx5
.0)1(
3
1
)ln(
2
3
22
=++ ydyxd'
eZ?￥Y31
.)1(
3
1
ln
2
3
22
Cyyx =++
VFE
è 5 p±sZ?
.
1
32
￥Y3 p±sZ?
x
yxx
dx
dy
+
++
=
3 1
? ?¤
,
1
1
2
xy
xdx
dy
=
+
+
"è
M^E
#
TE
.
43
43
C
xx
xyy =+++Y31
.
1 x
C
y
+
=?
QZ?Y31
.
1
)(
x
xC
y
+
=
!
.
43
)(
43
C
xx
xC +=
],[
1
1
2
1
1
Cdxexey
dx
x
dx
x
+
∫
∫
=
∫
++
è 6
~ ?yvD ?Dy
3 ? ?¤
,0)1()(
32
=++++ dyxdxyxx
,1
x
Q
y
P
==
Q,
^ ?±sZ?∴
"¨ wLsE
,)(),(
00
32
∫∫
+++=
yx
dydxyxxyxu
#X±sE
,0)(
32
=++++ dxxdxxydxxdydy
0
43
)(
43
=+++
x
d
x
dxyddy
.0)
43
(
43
=+++
xx
xyyd
~ ?yvD ?Dy
$??sE
,
32
yxx
x
u
++=
Q
∫
++∴ dxyxx )(
32
),(
43
43
yCxy
xx
+++=
),(yCx
y
u
′
+=
∴,1 x
y
u
+=
?
,1)( xyCx +=
′
+∴,1)( =
′
yC
,)( yyC =
eZ?￥Y31
.
43
43
C
xx
xyy =+++
~ ?yvD ?Dy
?a V?¨ú¨Z?
+?
.,,
)1(?
′
n
yyy L#?Ac??f
)(
)(
xfy
n
=
1a?
3E
,)(
)(
Q ??s| nxfy
n
= V¤Y3,
è 7
.cos
2
￥Y3 pZ? xey
x
=
′′′
3
1
2
sin
2
1
Cxey
x
+?=
′′
21
2
cos
4
1
CxCxey
x
+++=
′
32
2
1
2
2
1
sin
8
1
CxCxCxey
x
++++=
~ ?yvD ?Dy
è 
k p xy =
′′
￥üV? M
0 1 ON?D
°L 1
2
+=
x
y M M￥s wL,
3
2
1
,1
2
=
′
+= y
x
y
ù51 p3
2
1
,1,
00
=
′
==
′′
== xx
yyxy
,
2
1
1
2
Cxy +=
′
∴
} ?¤
2
1
0
=
′
=x
y
|C
2
1
1
=,
2
1
2
1
2
+=
′
∴ xy
,
2
1
6
1
2
3
Cxxy ++=
,1
0
=
=x
yQ
1
2
=∴C
1
2
1
6
1
3
++=∴ xxy
p wL1
~ ?yvD ?Dy
} ?eZ?,¤
3E
+?
.y?Ac??f
),(xpy =
′
7
,p
dx
dp
y
′
==
′′
5
).,( pxfp =
′
),,(
1
cxp?=
V¤eZ?Y3
),( yxfy
′
=
′′
2a?
p Y3
),(
1
cx
dx
dy
='
∫
+=
21
),( cdxcxy?
~ ?yvD ?Dy
è 9,0 ￥Y3 pZ? =
′
+
′′
yyx
3
,py =
′
7
p
dx
dp
y
′
==
′′
5
,0=+∴ p
dx
dp
x
x
dx
p
dp
= '
cxp +?= ||ln||ln
c
epx =||
x
c
p
1
=∴
)(
1
c
ec ±=
,
1
x
c
dx
dy
='
21
||ln cxcy +=∴
~ ?yvD ?Dy
)( ypy =
′
!
,
dy
dp
p
dx
dy
dy
dp
y =?=
′′
5
B¨Z?￥} ?eZ?¤??f
)( yp
p¤ 31
eZ?Y31
,
),(
2
1
Cx
Cy
dy
+=
∫
+?,x·
?Ac1M

3E
),,()(
1
Cyyp
dx
dy
==
),( yyfy
′
=
′′
3a?
),( pyf
dy
dp
p =
~ ?yvD ?Dy
.0
2
￥Y3 pZ? =
′
′′
yyy
3,
dy
dp
py =
′′
5
),(ypy =
′
!
} ?eZ?¤,0
2
= p
dy
dp
py,0)( = p
dy
dp
yp'
? 0= p
dy
dp
y
,
1
yCp = V¤
.
1
2
xC
eCy=
eZ?Y31
,
1
yC
dx
dy
=∴
è 10
y
dy
p
dp
=∴
Cyp +=∴ ||ln||ln
~ ?yvD ?Dy
è 11
3
.0',0,1
00
2
===
′
′′
== xx
yyyy p3Z?
,xNZ??Ac,y9?Ac
,,
dx
dp
ypy =
′′
=
′
5
7
¤} ?Z?,
,1
2
=+ p
dx
dp
.1
2
p
dx
dp
='
?
SHq 0
0
=
′
=x
y
01
2
≠? p?
dx
p
dp
=
∴
2
1
1
1
1
ln
2
1
Cx
p
p
+=
+
∴
~ ?yvD ?Dy
?
SHq 0
0
=
′
=x
y
,0
1
=C?
x
e
p
p
2
1
1
=
+
∴
1
1
2
2
+
==
′
x
x
e
e
py
chx
shx
=
2
ln Cchxy +=∴
?
SHq
0
0
=
=x
y
,0
2
=C?
chxy ln=∴
~ ?yvD ?Dy
1aB¨±sZ?l2
s ?M
E è
M^E ?±sZ?
B¨±sZ?
~ ?yvD ?Dy
± I5 1
Z?
0
32
4
22
3
=
+ dy
y
xy
dx
y
x
^?1 ?±sZ?$
~ ?yvD ?Dy
± I5 13s
=
3
2
y
x
yy
P
Q,
6
4
y
x
=

=
4
22
3
y
xy
xx
Q
,
6
4
y
x
=
x
Q
y
P
=
∴
eZ?
^ ?±sZ?,
~ ?yvD ?Dy
± I5 2
X? 3
1
=y 
2
2
3 xy += 
x
exy ++=
2
3
3
?
^±sZ?
()()()()162222
22
=?+
′

′′
xyxyxyxx
￥3 pNZ?
?
QZ?￥Y3,
~ ?yvD ?Dy
± I5 23s
321
,,yyyQ
?
^±sZ?￥3,
,
23
x
eyy =?∴
,
2
12
xyy =?
^?
QZ?￥3,
2
12
23
x
e
yy
yy
x
=
Q ≠è
pY31∴
.
2
21
xCeC
x
+=
( ) ( )
122231
yyCyyCy?+?=
~ ?yvD ?Dy
Ba
Y/
Z??
't
^ ?±sZ?
i p ?±sZ
?￥Y3
a 0)2( =?+ dyyxedxe
yy

a 0)(
22
=++ xydydxyx 
a 02)1(
22
=++ θρρ
θθ
dede 
=a ?¨43E p/
Z?￥sy0
i p Y
3
a 0
2
=+? xdxyxdyydx 
a dxyxydyxdx )(
22
+=+ 
a 0)1()1( =?++ xdyxyydxxy 
5B
~ ?yvD ?Dy
?a ￡
)]()([
1
xygxyfxy?
^±sZ?
 0)()( =+ dyxyxgdxxyyf ￥sy0
i pZ?
0)22()2(
2222
=?++ dyyxxdxyxy ￥Y3
1aX?
2
1
)0( =f
k ?? )(xf
P
0)()]([ =++ dyxfydxxfe
x
1 ?±sZ?
i pN
?±sZ?￥Y3
~ ?yvD ?Dy
5Bs?
Baa Cyxe
y
=?
2
 a?
^ ?±sZ?
a Ce =+ )1(
2θ
ρ 
=aa C
x
y
x
=+
2
2
 a
x
Ceyx
222
=+ 
a
xy
Ce
y
x
1
= 
?a
22
1
2 yx
eCyx =  C
yxy
x
=?+
222
1
1ln
1a Cyxexexf
xx
=++= )
2
1
(,)
2
1
()( 
~ ?yvD ?Dy
Ba p/
ò±sZ?￥Y3
a
x
xey =
′′′
 a
2
1 yy
′
+=
′′

a yyy
′
+
′
=
′′
3
)(  a 0
1
2
2
=
′
+
′′
y
y
y 
=a p/
ò±sZ?
@
ó
SHq￥+3
a 0,1,01
11
3
=
′
==+
′′
== xx
yyyy 
a 1,0,0
00
2
=
′
==
′
′′
== xx
yyyay 
a 2,1,3
00
=
′
==
′′
== xx
yyyy 
?a
k p xy =
′′
￥üV? )1,0(M ON?D°L
1
2
+=
x
y M M￥s wL
5=
~ ?yvD ?Dy
5=s?
Baa
32
1
2
3 CxCx
C
exey
xx
+++?= 
a
21
)cos(ln CCxy ++?= 
a
12
)arcsin( CeCy
x
+= 
a
xCxC
y
21
1
1
+
= 
=aa
2
2 xxy?=  a )1ln(
1
+?= ax
a
y 
a
4
)1
2
1
( += xy 
?a 1
2
1
6
1
3
++= xxy