제 3 강
2계 상미분방정식
1.2계 선형 상미분방정식(Linear D. E. of second order)
1. 2계 선형 상미분방정식(Linear D. E. of second order)
2. 다른 근 구하기
(How to obtain another basis if one basis is known)
' ' '
' '
' '
' '
' ,
' '
'
. ) (
) (
) (
) (
: 0 )
( '
) ( '
'
1 1
1 1
2 1
1 2
1 2
1 2
2 1
2 1
uy y
u y
u y
u y
uy y
u y
y u y
x y u
y
unknown y
known y
al proportion Not
y k y y
x q y
x p y
+ +
+
= +
=
×
=
=
®
¹
= +
+
2. 다른 근 구하기
(How to obtain another basis if one basis is known)
0 )
'
"
( ' ) '
2 (
"
) ' '
(
"
' ' 2
"
'
"
1 1
1 1
1 1
1 1
1 1
1 1
= +
+ +
+ +
×
=
+ +
+ +
+
= +
+
u qy py
y u
py y
u y
quy uy
y u p uy
y u y
u qy py
y
ò ò
ò
ò ò
ò ò
×
=
=
\
+ -
=
=
+ -
= +
-
= - +
=
¬
= +
+
×
=
=
= +
+
×
1 2
1 1
1 1 1
1 1
1 1
1 1
1
1 1
1
,
) ' )
( 2 ( exp '
) ' )
( 2 ( exp ,
' ) ( 2
ln
' 2
. 0
) '
2 ( '
'
"
, '
, 0 '
) '
2 (
"
y dx U y
dx U u
dx y p
U y u
dx y p
U y dx
y p U y
y dx py y
U dU
form Separable
U py y
U y
U u
U u
u py y
u
y
3. 2계 상수계수 선형 제차 상미분방정식(Second order
homogeneous ordinary D.E. with constant coefficients)
3. 2계 상수계수 선형 제차 상미분방정식(Second order
homogeneous ordinary D.E. with constant coefficients)
3. 2계 상수계수 선형 제차 상미분방정식(Second order
homogeneous ordinary D.E. with constant coefficients)
3. 2계 상수계수 선형 제차 상미분방정식(Second order
homogeneous ordinary D.E. with constant coefficients)
예제 1
sol
. 0
"
16 y - p
2y =
x x
h
x x
e c e
c y
e y
e y
y y
4 2 4
1
4 2
4 1
2 1
2 2
2
, , 4 4
0 4 )
( 4 ) 16 (
16 0
"
p p
p p
l p l p
l p l p
l p p
+
=
=
=
= -
=
= -
+
= -
= -
- -
3. 2계 상수계수 선형 제차 상미분방정식(Second order
homogeneous ordinary D.E. with constant coefficients)
예제 2
sol
. 0 '
2
" + ky + k
2y = y
kx kx kx
h
kx kx
e x c c
e x c e
c y
e x y
e y
k k
k
- - -
- -
+
=
× +
=
×
=
=
= +
= +
+
) (
,
0 )
( 2
2 1
2 1
2 1
2 2
2
l l
l
k
-l
= , 중근3. 2계 상수계수 선형 제차 상미분방정식(Second order
homogeneous ordinary D.E. with constant coefficients)
예제 3
sol
0 )
4 (
4
' 2'
'
+ y + + y =
y w
x h
x x
e x c
x c
y
x e
y x
e y
i i
i
2 2
1
2 2
2 1
2 1
2 2
2 2
) sin
cos (
sin ,
cos
2 ,
2
2 2 )
4 ( 4 2
0 )
4 ( 4
- -
-
+
=
\
×
=
×
=
- -
= +
-
=
± -
=
-
± -
= +
-
± -
=
= +
+ +
w w
w w
w l
w l
w w w
l
w l
l
3. 2계 상수계수 선형 제차 상미분방정식(Second order
homogeneous ordinary D.E. with constant coefficients)
예제 4
sol
2 6
) 0 ( ,
2 )
0 ( ,
0 )
1 4
(
2
' 2 '"
-
= -
=
= +
+
- y p y y y p
y
x x
x x x
h
x x
e x c
x y
c y
e x c
x c
y
x e
y x e
y
i i
i
) ) 2 sin 2
cos 2 ((
2 6
) 0 (
2 )
0 (
) 2 sin 2
cos (
2 sin ,
2 cos
2 1
, 2 1
2 1 4
1 ) 1 4
( 1 1
0 ) 1 4
( 2
' 0 2
'
1
2 1
2 1
2 1
2 2
2 2
p p
p p
p p
p p
p
p p
p p
p l
p l
p p
p l
p l
l
+ -
= -
=
= -
=
+
=
÷ ÷ ø ö ç ç
è æ
×
=
×
=
× -
=
× +
=
±
= -
±
= + -
±
=
= + +
-
=
4. 미분연산자(Differential operator)
' ' )
( )
( ,
'
,
22
2
y
dx y d dx
dy dx
D d D D
dx y Dy dy
dx
D = d = =
y=
y= = =
by ay
y
by aDy
y D
y b aD
D
y D P y
L
b aD
D D
P L
+ +
=
+ +
=
+ +
=
=
+ +
=
=
'
"
) (
) )(
( )
(
) (
2 2
2
4. 미분연산자(Differential operator)
예제 5
( 64 D
2+ 16 D + 1 ) y = 0 sol
x
x
y x e
e y
y y
y
y y
y
1 8 1 2
8 1 1
2 2
, 8 , 1
0 8 )
( 1 64
1 4
1
64 0 ' 1
4
" 1
0 '
16
"
64
-
-
= ×
=
-
=
= +
= +
+
= +
+
= +
+
l
l l
l
5. 2계 선형 제차 상미분방정식 / Euler-Cauchy equation
0 '
"
2
y + axy + by =
x
0 )
1 (
0 )
1 (
, 0
0 }
) 1 (
{
) 1 (
) 1 (
) 1 (
"
, '
,
2
1 2
2
2 1
= +
- +
= +
+ -
¹
= +
+ -
=
+ +
-
=
+
× +
-
×
-
=
=
=
- -
- -
b m
a m
b am
m m x
x b am
m m
bx amx
x m
m
bx mx
ax x
m m x
x m
m y
mx y
x y
m
m m m
m
m m
m
m m
m
: m의 특성 방정식
5. 2계 선형 제차 상미분방정식 / Euler-Cauchy equation
0 )
1
2
(
= +
-
+ a m b
m
24 )
1 (
) 1
( 2
2 , 1
b a
m - a - ± - -
=
m m
h
m m
x x
u y
x u y
y k y
m m
m root
double real
b a
ii
x c x
c y
x y
x y
m m
roots real
distinct
b a
i
) ln
) ( ( )
( ,
)
; (
, 0 4
) 1 (
)
,
) ,
; (
, 0 4
) 1 (
)
1 2
1 2
2 1
2
2 1
2 1
2 1
2
2 1
2 1
=
×
=
¹
=
=
= -
-
+
=
=
=
>
-
-
5. 2계 선형 제차 상미분방정식 / Euler-Cauchy equation
예제 6
10 x
2y " + 46 xy ' + 32 . 4 y = 0
sol
x x
c x
c y
x x
y x
y m
m m
m m
m
h
ln
ln ,
5 , 9
10 0 18
10
324 18
18
0 4
. 32 36
10
4 . 32 46
) 1 (
10
5 9 2
5 9 1
5 9 2
5 9 1
2 2
× +
=
×
=
=
-
=
±
= - -
±
= -
= +
+
=
+ +
-
- -
- -
중근
5. 2계 선형 제차 상미분방정식 / Euler-Cauchy equation
예제 7
sol
0 )
( xD
2+ D y =
x c
c y
x y
x y
m m
m m
m
equation Cauchy
Euler xy
y x
y xy
h
ln
ln ,
1
, 0 ,
0 )
1 (
: 0 '
"
0 '
"
2 1
2 0
1
2 2
+
=
=
=
=
=
=
= +
-
-
= +
=
+
6. 2계 선형 비제차 상미분방정식
미정 계수법(method of undetermined coefficients)
) ( )
( '
) (
"
p x y q x y r x
y
+ + =일반해 : (여기서 y
y
=y
h +y
p p 는 r(x)의 근)) ( '
" ay by r x
y + + =
6. 2계 선형 비제차 상미분방정식
미정 계수법(method of undetermined coefficients)
rx rx
n n
n n n
rx rx
p
e wx M
wx K
wx ke
wx M
wx K
wx k
wx M
wx K
wx k
k x
k x
k x
k n
kx
ce ke
y for Choice
x r in Term
) sin
cos (
cos
sin cos
sin
sin cos
cos
) 3 , 2 , 1 , 0 (
) (
0 1
1 1
1
+
×
+ +
+ +
+ +
= L
- -L
6. 2계 선형 비제차 상미분방정식
예제 8
y " - y = 2 e
x+ 6 e
2 xsol
x x
x x
x x
p
x x
x x
x x
x x
x
x x
x x
p
x x
x p
x x
x p
x x
e xe
y
k k
k k
e e
e k e
k
e k xe
k e
k xe
k e
k
e k xe
k e
k e
k y
e k xe
k e
k y
e e
k xe
k y
e y
e y
2 2
2 2
1 1
2 2
2 1
2 2 1
2 2 1
1
2 2 1
1 1
2 2 1
1
2 2 1
2 1
2 , 1 2
2
2 ,
6 3
1 ,
2 2
6 2
3 2
4 2
4
"
2 '
) (
, 1 , 1
0 )
1 (
) 1 (
1
+
=
=
=
=
=
+
= +
=
- -
+ +
+ +
+
=
+ +
=
+
=
=
= -
=
= -
+
= -
- -
중근 가
실근 l 두
l l
l
6. 2계 선형 비제차 상미분방정식
변수변환법(Method of variation of parameters)
) ( )
( '
) (
" p x y q x y r x
y + + =
. ,
)
( x 에 대한 근 y p 를 구하는 가장 일반적 해법이다 r
W dx r y y
W dx r y y
x
y
p( ) = -
1ò
2× +
2ò
1×
W dx r y W
W dx r y W
x y
or p ( ) = 1
ò
1 × + 2ò
2 ×6. 2계 선형 비제차 상미분방정식
변수변환법(Method of variation of parameters)
증명 대한
의 y p 에 x
r qy
py
y " + ' + = ( )
6. 2계 선형 비제차 상미분방정식
예제 9
y " - 4 y ' + 5 y = e
2 x× cosec x sol
dx x x
e x
x e
e dx e x x e
x e
e dx e x x e
x e
y
e x
x e
x e
x e
x e
x e
x e
x e
x e
x e
x e
x e
x e
x W e
x e
y x e
y
i
x x
x x x
x x
x x
x p
x x
x x
x x
x x
x x
x x
x x
x x
cot sin
cos
sin cos 1
sin sin sin 1
cos
) sin (cos
) sin cos
2 ( sin )
cos sin
2 ( cos
cos sin
2 , sin cos
2
sin ,
cos
sin ,
cos
, 2
5 4 2
0 5 4
2 2
4 2 2
2 4
2 2
2
4 2
2 4
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 1
2
× +
×
× -
=
×
×
×
× +
×
×
×
× -
=
= +
=
× -
×
× -
× +
×
×
=
× +
×
× -
×
×
= ×
×
=
×
=
±
= -
±
=
= + -
ò
ò ò
공액복소근 l 두
l l