제 4 강
고계 상미분방정식
1. 고계 선형 미분방정식
) ( )
( )
( )
( )
(
( 1 ) 2 ( 2 ) 1 ' 01 )
(
p x y p x y p x y p x y r x
y
n+
n - n -+
n - n -+ L + + =
y n
y
y 1 , 2 , L,
2. Wronskian, W
Wronski 판별법 ( Wronski determinant)
제차 미분방정식의 일반해, y h 의 기본근의 비례 여부 판별법
정행렬을 사용하여, W으로 표현
W = 0 : 비례한다.
W ≠ 0 : 각각의 기본근은 비례하지 않는다.
' 1 2 '
2 ' 1
2 '
1
2 1
2 1
, )
( y y y y
y y
y y y
y
W = = -
y 1 , y 2 의 경우
2. Wronskian, W
Wronskian, W (y 1 , y 2 , y 3 기본근의 3계미분방정식)
' 1 2 '
2 1 '
' 2 ' ' 1 ' 1
2 1
' ' 2 ' ' 1
' 2 ' 1
2 1
3
' 3 1 '
1 3 '
' 1 ' 1 1
' ' 3 '
' 1
' 3 '
1
3 1
2
' 2 3 '
3 2 '
' 2
2
' ' 3 ' ' 2
' 3 ' 2
3 2
1
' ' 2 ' ' 1 ' 1
2 1
' ' 3 ' ' 2 ' ' 1
' 3 ' 2 ' 1
3 2 1
1 0 0
1 0
1 0 0
1 1
1 0 0
1 0 0
y y y
y y
y y
y y
y y
y y
y y W
y y y
y y
y y
y y
y y
y y
W
y y y
y y
y
y y
y y
y y W
y y y
y y
y y y
y y y
y y y W
× -
×
=
=
× -
×
=
=
×
× -
×
×
=
=
=
=
' ' 3 ' 1 2 ' ' 2 ' 3 1 ' ' 1 ' 2 3
' ' 2 ' 1 3 ' ' 1 ' 3 2 ' ' 3 ' 2 1
y y y y y y y y y
y y y y y y y y y
×
× -
×
× -
×
× -
×
× +
×
× +
×
×
3. 상수계수 고계 제차 미분방정식(Higher order homogeneous D. E. with constant coefficients)
x x
e y
x x
e y
x e
y x
e y
double
x e
y x
e y
simple
multiple) (simple,
roots complex
: ΙΙΙ case iii
e x c xe
c e
c y
e x y
, xe y
, e y
roots real
Mutiple :
ΙΙ case ii
e c e
c e
c e
c y
roots real
District :
Ι case i
a a
a a
y a y
a y
a y
a y
ax ax
ax x
x x
x x
x h
λx λx
λx
x n x
n x
x h
n n
n n
n n
n n
n n
n
n n
b b
b b
b b
l l
l l
l l
l l l l
l l
a
a a
l l
l
l l
l l
sin ,
cos
sin ,
cos :
sin ,
cos :
)
) (
, )
) , , , ,
( . )
0 0
4 3
2 1
2 1
2 3 2
1
2 3
2 1
3 2
1 1 2
1
3 2 1
0 1
2 2 1
1
0 '
1 )
2 ( 2 )
1 ( 1 )
(
1 2
1
×
×
=
×
×
=
×
=
×
=
×
=
×
=
+ +
=
=
=
=
=
=
= + +
+ +
=
= +
+ +
+ +
= +
+ +
+ +
-
- -
- -
-
- -
- -
L
L L
L
(u(x)=x가 중첩적용)
(u(x)=x)
3. 상수계수 고계 제차 미분방정식(Higher order homogeneous D. E. with constant coefficients)
예제 1 ( 16 D 4 - 40 D 2 + 9 ) y = 0
sol
x x
x x
h
x x
x x
IV
e c e
c e
c e
c y
e y
e y
e y
e y
y y
y
2 3 4 2
3 3 2
1 2 2
1 1
2 3 4
2 3 3
2 1 2
2 1 1
4 , 3 2
2 , 1 2
2 2
2 4
"
, ,
,
2 , 3
3 2
, 0 9 )
2 (
2 , 1
1 2
, 0 1 )
2 (
0 ) 9 4
)(
1 4
( 9 40
16
0 9
40 16
- -
- -
+ +
+
=
=
=
=
=
±
=
±
=
= -
±
=
±
=
= -
= -
-
= + -
= +
-
l l
l
l l
l
l l
l l
3. 상수계수 고계 제차 미분방정식(Higher order homogeneous D. E. with constant coefficients)
예제 2
sol
0 16 =
- y
y
IVx c
x c
e c e
c y
x y
x y
e y
e y
i
x x
h
x x
2 sin 2
cos
2 sin ,
2 cos ,
,
2 ,
2
0 )
4 )(
4 (
16
4 3
2 2 2
1
4 3
2 2
2 1
4 , 3 2
, 1
2 2
4
+ +
+
=
=
=
=
=
±
=
±
=
= -
+
= -
- -
l l
l l
l
4. 고계 비제차 미분방정식
(Higher order nonhomogeneous D. E)
4. 고계 비제차 미분방정식
(Higher order nonhomogeneous D. E)
예제 3
x
3y
' ' '+ x
2y
' '- 2 xy
'+ 2 y = x
3ln x sol
2 3 2
1 1
2 3
2 1
1
2
2 2
3
2 2
3
, ,
2 , 1 , 1
0 )
1 (
) 1 (
) 2 (
) 1 (
) 2 (
) 2 (
) 2 (
2 2
2 2
2 3
2 2
) 1 (
) 2 )(
1 (
x c x
c x
c y
x y
x y
x y
m
m m
m m
m
m m
m m
m m
m m
m m
m m
m m
m m
m m
h
= + +
=
=
=
-
=
= +
- -
= -
-
=
- -
-
= +
- -
=
+ -
- +
+ -
=
+ -
- +
- -
- -
4. 고계 비제차 미분방정식
(Higher order nonhomogeneous D. E)
예제 3
x
3y
' ' '+ x
2y
' '- 2 xy
'+ 2 y = x
3ln x
1 1
1 3
2 1
3 2 1
3
3 2 1
3 2
2 1
2
2 2
2 2
1
1 1
1 1
1 3
2 1
3 2
2 1
2 0
2 1
0 2
0 1
0
3 2
1 1
2
0
2 1 2
2 0 0
2 0
1 0 0
2 0 1
2 1 0 0
6 2
2 4
2 0
2 2
0 2
2 1
- -
- -
- -
- - -
- - -
- - -
- -
- -
- -
- -
- - -
= +
= -
-
=
-
= - -
= -
-
=
= -
=
=
= +
- +
= -
-
=
x x
x x
x
x x
x x
x x
W
x x x
x
x x
x x
W
x x
x x
x x x W
x x
x x
x x
x
x x
x
x x
x x x
W
x x
r ( ) = ln
4. 고계 비제차 미분방정식
(Higher order nonhomogeneous D. E)
예제 3
x
3y
' ' '+ x
2y
' '- 2 xy
'+ 2 y = x
3ln x
[
3 3
2 3 2
1 1
3 3
3 3
3 3
3 3
2
2 2
4 4
2 3
1
1 1 2
1 1
2 1
32 ln 7
8 1
32 ln 7
8 1 3
ln 1 3 1 8
ln 1 4 1 96
ln 1 24
1
ln 1 3
1 2 ln 1
2 1 2 1
4 ln 1
4 1 6
1
3 ln 2 ln
6 ln
6 ln 2
6 ln 3 6
ln
x x
x x
c x c x c y
x x
x x
x x x
x x x
x x
x dx x x
x x
x dx x x
x x x dx
x x
x x
dx x x
dx x x x
dx x x x
x dx x x x
x dx x x
x dx x x x
y p
-
× +
+ +
=
-
= -
+ +
- -
×
=
ú û
× ù -
× +
ú û
× ù ê ë
é × - ú -
û
× ù ê ë
é × -
=
+
× -
×
=
+ ×
× + -
= ×
- -
- - -
- -
ò
ò ò
ò ò
ò
ò ò
ò
5. 부분 분수(Partial fractions)
) (
) ) (
( g x
x x f
y =
, ) (x
g of type to
According
b x
E a
x a x
D Cx a
x a x
B Ax
b x a
x a x x
y
a x a x factors complex
Repeated IV
Case iv
b x
C a
x a x
B Ax b
x a x a x x
y
a x a x factor Complex
III Case iii
b x
D a
x C a
x B a
x A b
x a x x
y
a x factors Repeated
II Case ii
b x
B a
x A b
x a x x
y
b x a x factors unrepeated
I Case i
m
+ - -
- + +
- -
= +
- -
= -
- -
+ - -
-
= + -
-
= -
- -
+ - + -
+ -
= - -
= -
-
+ -
= - -
= -
- -
)]
( ) [(
)]
( ) [(
) (
)]
( ) [(
) 1 (
)]
( ) [(
: )
) (
) (
) (
) (
) (
) 1 (
, ) (
) (
: )
) (
) (
) (
) (
) 1 (
, ) (
: )
) (
) (
) 1 (
, ) (
) (
: )
2 2
2 2 3
3
5. 부분 분수(Partial fractions)
예제 4 ( 3 )( 2 ) 1
- +
= +
x x
x y x
sol
) 2 (
10 3 )
3 (
15 2 6
1
10 , 3
15 , 2
6 , 1
2 3
) 2 )(
3 (
1
+ - - +
-
=
= -
= -
- = + +
+
=
- +
= +
x x
x
C B
x A C x
B x
A
x x
x y x
5. 부분 분수(Partial fractions)
예제 5
sol
) 2 )(
1 (
4 4
2
2 3
- -
+
= -
x x
x
x y x
2 1 1
1 3
2
1 ,
1 ,
3 ,
2
2 1
) 2 )(
1 (
4 4
2 2 2
2 3
- - - -
+
=
-
= -
=
=
=
+ - + -
+
=
- -
+
= -
x x
x x
D C
B A
x D x
C x
B x
A
x x
x
x
y x
5. 부분 분수(Partial fractions)
예제 6
sol
) 1 (
2 ) 2
( 3 2
2 3
+ +
= +
x x
x x x
y
) ( x y
5. 부분 분수(Partial fractions)
예제 7
sol
1 ) 2
( 2
= + x x x
y
