:

-

2005/9/23 W. Y. Han 2

Linear Ordinary Differential

Equationn

ydegree1y

1.51.6

),()()()()()( 122

21

1

10 xbyxadxdyxa

dxydxa

dxydxa

dxydxa nnn

n

n

n

n

n

=+++++

L

6.1

5.1065

1.10

33

32

4

4

2

2

2

2

2

xxedxdyx

dxydx

dxyd

ydxdy

dxyd

dxdyxy

dxyd

=++

=++

=

+

1.1degree21.51.6

2

dxdy

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2005/9/23 W. Y. Han 4

()

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()

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()

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()

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()

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()

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()

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()

yyyy

yy

21

21

21

,,

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()

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()

2005/9/23 W. Y. Han 14

)2()1(,,

)1(0

LL

LLL

xeyba

byyay

=

=++

)4(21),4(

21

,)1()3()3(0)(0)(

)1(

22

21

22

2

baabaa

baeba

eyey

x

xx

=+=

=++=++

==

LL

2005/9/23 W. Y. Han 15

:

)4(21),4(

21

22

21

baabaa =+=

04,:304,:2

04,:1

2

2

2

baba

ba

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()

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()0=++ byyay

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()

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()0=++ byyay

y1(method of reduction order) y2 (p75)

=

=

Udxyy

ey

UPdx

12

21

1

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()0=++ byyay

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())1(0 LLL=++ byyay

:

)4(21),4(

21

22

21

baabaa =+=

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()

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()

)1(0 LLL=++ byyay

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-

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-()

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-()

025.2: 2 = yyxyx

025.30)2()15.2(

2,5.2

2

2

=+

=++

==

mmmm

ba

5.0,40)5.0)(4(

==+

mmm xx

xxy 1121

215.0

1 ====

42 xy =

4212211

1 xcx

cycycy +=+=

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-()

2/)1(1:

axyII =y1(method of reduction order) y2 (p95)

xxxyy a lnln 2/)1(12==

2/)1(21

2/)1(2

2/)1(12211

)ln(:

lna

aa

xxccy

xxcxcycycy

+=

+=+=

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2/)1(21 )ln(:

axxccy +=

043: 2 =+ yyxyx

04404)13(

4,3

2

2

=

=++

==

mmmm

ba

2,20)2)(2( == mmm

xxxyyxy lnln, 2122

1 ===

2212211 )ln( xxccycycy +=+=

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-()

)]lnsin()lncos([

,:

21

xvBxvAxy

ivumivumIII

u +=

=+=

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)]lnsin()lncos([

, 21xvBxvAxy

ivumivumu +=

=+=

0137: 2 =++ yyxyx

0136013)17(

13,7

2

2

=++

=++

==

mmmm

ba

im 2313932,1 ==

2,3 == vu

[ ])ln2sin()ln2cos(3 xBxAxy +=