Engineering Mathematics I

(1)

Bong-Kee Lee

School of Mechanical Systems Engineering Chonnam National University

Engineering Mathematics I

3. Higher Order Linear ODEs

3.1 Homogeneous Linear ODEs

 고계 선형상미분방정식

– n계(n-th order) 상미분방정식: 미지의 함수의 n차 도함수

가 최고차 도함수 항으로 나타날 때

– n계 선형상미분방정식

• 표준형(standard form)

 

n n n

dx y y d

dx y dy





2 2

''

  '



^x^,^y^,^y^'^, ^,^yⁿ



^⁰

F 

  p

 

x y^{ } p

 

xy p

 

x y r

 

x yⁿ  _n_₁ ⁿ^¹  ₁ ' ₀ 

nonhomogeneous homogeneous

 

x 0 r

 

x ⁰ r

(2)

School of Mechanical Systems Engineering Engineering Mathematics I

3.1 Homogeneous Linear ODEs

 제차 선형상미분방정식

– 기본 정리: 중첩(superposition) 또는 선형(linearity) 원리

• 제차 선형상미분방정식에 대한 어떤 열린 구간에서 해들의 합과 상수 곱은 다시 주어진 구간에서 제차 선형상미분방정식의 해가 됨(비제차 선형방정식 혹은 비선형 방정식에서는 성립하지 않음)

– 일반해(general solution)

 

x cy cnyn

y  ₁ ₁ general solution

yn

y y₁, ₂,, basis or

fundamental system particular solution initial conditions

cn

c c₁, ₂,,

3.1 Homogeneous Linear ODEs

– 1차 독립(linearly independent)

   

0 0

2 1 1 1









n n n

k k k

x y k x y k



 

 

x x x x

x x

x iv x x

iv

e c e c e c e c y

e y e y e y e y

e e

e y e y e y

y y y

2 4 3 2 2 1

2 4

2 4 2

4

4 2

, , ,

2 , 1 , 1 , 2 0

4 5

0 4 5 4

5

, '' assumption

0 4 '' 5 example



















































(3)

3.1 Homogeneous Linear ODEs

 초기값 문제(initial value problem)

– 초기값 문제에 대한 존재성과 유일성 정리

 p_₁

 

x y^{ }^¹ p₁

 

x y'p₀

 

x y0

yⁿ _n ⁿ 

   

^{ }

 

0 1

1 1 0 0

0 K ,y' x K, ,yⁿ^ x K_n_ x

y  initial conditions

   

 

^on^.

solution unique a has problem value initial then the , in is and

interval open some on continuous are

ts coefficien the

If

0

1 0

I x y I

x

I x

,p , x p  _n

3.1 Homogeneous Linear ODEs

 초기값 문제(initial value problem)

– (3계 오일러-코시 방정식)

       

 

    

    

 

       

 

   

 

3 2 3

2 1 3

2 3 2 1

3 2 1

2 3 2 2 3 2 1

3 3 2 2 1

2

1 2

2 3 3

3 2

1 2 3

2 1 , 1 , 2 4

6 2 1 ''

1 3 2 1 '

2 1

6 2 '' , 3 2 ' step 2nd

3 , 2 , 1 0 3 2 1 6 5 1

0 6 6 1 3 2 1

6 6

1 3

2 1

2 1 ''

' , 1 ''

, '

that assume step 1st

4 1 '' , 1 1 ' , 2 1 with 0 6 ' 6 '' 3 '' ' example

x x x y c

c c c

c y

c c c y

x c c y x c x c c y

x c x c x c y

m m

m m m m m

x m m m m m m

x mx x x m m x x m m m x

x m m m y x m m y mx y

x y

y y y y

xy y x y x

m

m m m

m

m m

m





















































































































general solution

particular solution initial conditions

(4)

3.1 Homogeneous Linear ODEs

 Wronskian 또는 Wronski 행렬식

– 해의 1차 종속과 1차 독립

 

   1  1

2 1 1

2 1

1

' '

, ' ,





n n n

n

n n

n

y y

y

y y

y

y y

y y y W











   

. on solutions of

basis a form that they so

, on t independen

linearly are then

zero, not is at which in

an is there if Here . on zero

y identicall is

then , for zero is if e, Furthermor .

in some

for zero is

nskian their Wro if

only and if on dependent linearly

are on solutions

Then

. interval open an on ts

coefficien continuous

have ODE Let the

1 1

0 0

1

1 0

I I

,y , y W

I x I

W x x W

I x x

I I

,y , y n

I x

,p , x p

n n

n







3.1 Homogeneous Linear ODEs

– 일반해의 존재성

– 일반해는 모든 해를 포함한다

   

. on solution general

a has ODE then the ,

interval open some on continuous are

ts coefficien the

If ₀ ₁

I

I x

,p , x p  _n

   

       

constants.

suitable are and

on ODE the of solutions of

basis a is where

form the of is on solution

every then ,

interval open some on ts

coefficien continuous

has ODE the If

1 1

1 0

n n

n

,C , C I ,y

, y

x y C x y C x Y

I x Y y

I x

,p , x p









(5)

3.2 Homogeneous Linear ODEs with Const. Coffs.

 상수계수를 갖는 제차 선형상미분방정식

 p_₁

 

x y^{ }^¹ p₁

 

x y'p₀

 

x y0

yⁿ _n ⁿ 

   

0 ' ₀

1 1

1    

a_y ^ ay a y

yⁿ _n ⁿ 

 ¹ ₁ ₀ 0

1    

a_n_ ⁿ^ a a

n  

  characteristic equation

ex

y ^

3.2 Homogeneous Linear ODEs with Const. Coffs.

– 서로 다른 실근

– 단순 복소근

– 다중 실근

– 다중 복소근

x n x x

n

en

y e y e

y ^ ^ ^



 ₁, ₂,,  ₁ ¹, ₂ ²,, 

x e y x e y

i ^x  ^x 



   ₁ ^ cos , ₂ ^ sin

x m x x

x xe xe x e

e

m ^ ^ ^ ^

:realroot oforder  , , ² ,, ^¹

x xe x xe x e x e

i ^x  ^x  ^x  ^x 



  :complex doubleroot ^ cos , ^ sin , ^ cos , ^ sin

(6)

3.2 Homogeneous Linear ODEs with Const. Coffs.

 

 

       

x x x

e c e c e c y

e y e y e y

y y y y

2 3 2 1

2

2 3

, ,

2 , 1 , 1

0 2 1 1 2 1

0 2 2

equation stic characteri

0 2 ' '' 2 '' ' example













































general solution

3.2 Homogeneous Linear ODEs with Const. Coffs.

       

   

 

x x e

y

B A c A

c y

B c y

A c y

x B x A e c y

x B x A e c x y

x e i

i

y y

y y y

x x x

x x

10 sin 10 cos 3

1 , 3 , 1 299

100 0

''

11 10 0

'

4 0

10 sin 100 10 cos 100 ''

10 cos 10 10 sin 10 '

10 sin 10 10 cos

sin , 10 10 cos 1

10 , 1 0 100 1

0 100 100

299 0 '' , 11 0 ' , 4 0 with 0 100 ' 100 '' '' ' example

1 1

1 2

2 3

































































































general solution

particular solution

characteristic equation

(7)

3.2 Homogeneous Linear ODEs with Const. Coffs.

 

   

 

^x

x x x

x x

x iv v

e x c x c c x c c y

e x c xe c e c x c c y

e x y xe y e y

x y y

y y y y

2 5 4 3 2 1

2 5 4 3 5 4 3

2 1 2 1

5 4 3 2 1

2 3 2

3 2

2 3 4 5

or

, , : 1 2

, 1 : 0 1

1 ,

0

0 1 1

3 3

0 3

3

0 '' '' ' 3 3 example

















































general solution characteristic equation

3.3 Nonhomogeneous Linear ODEs

 n계 비제차 선형 상미분방정식

 p_₁

 

x y^{ }^¹  p₁

 

xy'p₀

 

x yr

 

x 0

yⁿ _n ⁿ 

 

x y

 

x y

 

x

y  _h  _p ^{ } ^ ^

n n h

n n n

y c y c y

y p y p y

p y











 _ ^



1 1

0 1 1

1 ' 0

   

p n n n

y

r y p y p y

p y







 _₁ ^¹  ₁ ' ₀ general solution of homogeneous ODE

any solution of

nonhomogeneous ODE (i) method of undetermined coefficients

(ii) method of variation of parameters general solution of nonhomogeneous ODE

(8)

3.3 Nonhomogeneous Linear ODEs

– 미정계수법(method of undetermined coefficients)

• 기본 규칙, 변형 규칙, 합 규칙

       

 

   

     

x p p p p

x x

x p

x x

x p

x x

x p

x p x x

h x x

x x

x p

h

x

e y y y y

e x x x C e x x x C e x x C y

e x x x C e x x C e x x C y

e x x C e x e x C y

e Cx y ke e x r

y e x c x c c e x c xe c e c y

e x y xe y e y y

y y

y e

y y y y



































































































30 ' 3 '' 3 '' '

9 18 6 6

6 3

12 6 '' '

6 6 3

3 6 ''

3 3

'

~ 30 step

2nd

, ,

1 0

1

0 1 3 3 step 1st :

47 0 '' , 3 0 ' , 3 0 with 30

' 3 '' 3 '' ' example

3 2 3

2 2

3 2 3

2 2

3 2 3

2

3 2 3 2 1 2 3 2 1

2 3 2

1 3

, 2 , 1 3

2 3



 characteristic equation

3.3 Nonhomogeneous Linear ODEs

– 미정계수법(method of undetermined coefficients)

     

 

       

 

     

   

^x

x x

x p

h

x p x

x

x x x x

x x p p p p

e x x y

c c

y

e x x c x c e

x x c c

y

c c

y e x x c x c e x x c c y

c y

e x x c x c c e x e x c x c c y y y

e x y C e e C

e e Cx e x x C e x x x C e x x x C

e y y y y

















































































































3 2

3 3

3 2 3 3

2 3 3

2 2

3 2 3 2 2

3 2

1

3 2 3 2 1 3 2

3 2 1

3

3 3

2 3

2

5 25 3

25 47

3 2 0 ''

5 15

2 3 15

15 2 2 ''

0 3 3 0 ' 5

3 15

2 '

3 0

step 3rd

5 5

30 6

30 3

3 6

6 3 9

18 6

30 ' 3 '' 3 '' '

 

x y

 

x y_p general solution

particular solution

(9)

3.3 Nonhomogeneous Linear ODEs

– 매개변수변환법(method of variation of parameters)

     

   

       

   



 





dx x x r W

x x W y dx x x r W

x x W y

dx x x r W

x x W y x y

n n m

k

k k p



1 1

1

  



0 0 0 1



.

tor column vec the

by of column th - j the replacing by :

, , 2 , 1

T j

W n

j x W



 

1 ' , 0 ' 1 0 '

' instance, for

1 1 2 2 2 1 2 1

2 1

y W y y W y y y

y

W y   

3.3 Nonhomogeneous Linear ODEs

– 매개변수변환법(method of variation of parameters)

 

      

       

 

2 2

3 3 2 3

2 4 2 3 2

1 3 2 3 2

3 3 2 2 1 1

3 3 2 2 1

3 3 2 2 1 2

4 2

3

1 2 0

0 2 1

0 ,

2 6 1 0

3 0 1

0 ,

6 2 1

3 2 0 0 , 2 6 2 0

3 2 1 step 2nd

, , 3

, 2 , 1 0 3 2 1 6 5 1

0 6 6 1 3 2 1 step

1st :

equation Cauchy - Euler : ln 6 ' 6 '' 3 '' ' example

x x x x W x x x x x W x x x x

x x W x x x x

x x x W

W rdx y W W rdx y W W rdx y W y

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 x y

y y y

x x y xy y x y x

p h

m p h





























































(10)

3.3 Nonhomogeneous Linear ODEs

– 매개변수변환법(method of variation of parameters)

 

 



 



 













 



 









 



 





 



 









 































6 ln 11 6 1

6 ln 11 6 ln 1

2 ln 4

2 ln 9

3 2

2ln ln 1

2 ln 1

2 ln 2 ln

ln 2 2

6 ln 6 ' 3 '' '' ' ln 6 ' 6 '' 3 '' '

step 3rd

4 3 3 2 2 1

4 3

2 2 2 3 3

3 2

2

3 2 3 3

3 2 3

4

3 2 4

2 3

3 3 2 2 1 1

x x x c x c x c y y y

x x x x x x x x

x x x x

x x

xdx x

xdx x x xdx x x

xdx x x x x xdx x x

x x xdx x x x x y

x r x x x y x y xy y x x y xy y x y x

W rdx y W W rdx y W W rdx y W y

p h p

p

3.3 Nonhomogeneous Linear ODEs

 Cofactor Expansion (Chap. 7.7)

 

. of column and row deleting by formed is

. det 1

: sign correct with the of

t determinan the

is The

. det

: row of cofactors the

and row of n combinatio a

is of t determinan The

2 2 1 1

A j i

M

M A

cofactor

A a A a A a A

i i

A

ij

ij j i ij

ij ij

in in i

i i i

 





 

32 31

22 21 13 33 31

23 21 12 33 32

23 22 11 33 32 31

23 22 21

13 12 11

a a

a a a a a

a a a a a a

a a a









² ²

 

³ ³



³ ³ ³

2 3 2 3 2 2 2

3 2

2 4 6 2 6 6 12

3 0 2 6 1 2 6 2

3 2 6 2 0

3 2 1

x x x x x x x x

x x

x x x x x x x x x x x x

x x x W























(11)

3.3 Nonhomogeneous Linear ODEs

 응용: 탄성보(elastic beams)

 

  EIy  f

 

x dx

x y

EId _iv ^iv

iv  

E: Young’s modulus of elasticity

I: moment of inertia of the cross-section about the z-axis f(x): load per unit length

 

^x

f

boundary conditions

   

 

y y x y y x L

L x

y y

L x

y y



at 0 '' ' '' and

&

0 at 0 ' C

and 0 at 0 ' B

and 0 at 0 '' A