Engineering Mathematics I

Bong-Kee Lee

School of Mechanical Systems Engineering Chonnam National University

Engineering Mathematics I

5. Series Solutions of ODEs. Special Functions

5.1 Power Series Method

 n계 선형상미분방정식

  p

 

x y^{ } p

 

xy p

 

x y r

 

x yⁿ  _n1 ^n1  ₁ ' ₀ 

  a y  ay ay r

 

x

yⁿ  _n1 ^n1 ₁ ' ₀ 

 ¹ _{1 0} 0

1    

a_n ^n a a

n  

 

characteristic equation ex

y ^

 

x y

 

x y

 

x y  _h  _p

general solution of nonhomogeneous ODE

 

x cy cnyn

y  _{1 1} general solution of homogeneous ODE

constant coefficients

School of Mechanical Systems Engineering Engineering Mathematics I

5.1 Power Series Method

 거듭제곱급수(멱급수, power series) 해법

– 변수계수를 가지는 선형상미분방정식을 풀이하는 표준해 법

• 해의 수치적 값을 계산하거나 특성을 조사하는데 사용됨

• 해에 대한 다른 표현을 유도하는데 사용됨

 거듭제곱급수

– x – x

₀

의 거듭제곱을 가지는 무한급수의 형태

– x

₀

= 0 인 경우







 



















^



2 0 2 0 1 0 0

0 a a x x a x x

x x a

m

m m

coefficient center









^ 



2 2 1 0 0

x a x a a x a

m m m

5.1 Power Series Method

 거듭제곱급수

– 거듭제곱급수의 예: Maclaurin 급수

   

   

^















 

 









 



























 











































! 5

! 3

! 1 2 sin 1

! 4

! 1 2

! 2 cos 1

! 3

! 1 2

! 1 1

1

5 3

0

1 2

2 2

0 2

3 2

0

2 0

2 2 1 0 0

x x x m x x

x x m

x x

x x x m e x

x x x x

x a x a a x a

m m m m

m m m

m x

m m m

m m

School of Mechanical Systems Engineering Engineering Mathematics I

5.1 Power Series Method

 거듭제곱급수 해법의 개념

   

     

 

 

   

m m

m m m

m m

m m m

a

y x q y x p y y y y

x a a x a m m y

x a x a a x ma y

x a x a x a a x a y x q x p

y x q y x p y

obtain

0 '

'' , ' , ''

2 3 2 1

''

&

3 2 '

&

solution

series power

~ ,

0 '

''

3 2 2

2

2 3 2 1 1

1

3 3 2 2 1 0 0





















































































5.1 Power Series Method

 거듭제곱급수 해법의 개념

 

   

2 0 6

4 2 0

0 6 0 4 2 0 0 3 3 2 2 1 0

0 2 6 0 2 4 0 2

2 6 2 4 0 2 5

3 1

3 5 2 4 1 3 0 2 1

4 3 3 2 2 1 0

3 3 2 2 1 0 2

3 2 1

2 3 2 1 1

1 3

3 2 2 1 0 0

! 3

! 1 2

! 3

! 2

!, 3 , 3

! 2 , 2

3, 2, ,

&

0

2 5 , 2 4 , 2 3 , 2 2 , 0

2 2 2 2

2 3

2

3 2 '

2 ' example

x

m m m m

m m

e x a

x x a

a x a x x a a x a x a x a a y

a a a

a a a

a a

a a a a a a a

a a

a a a a a a a a a

x a x a x a x a

x a x a x a a x x

a x a a

x a x a a x ma y x

a x a x a a x a y

xy y





 



    



























































































































^



 



























School of Mechanical Systems Engineering Engineering Mathematics I

5.1 Power Series Method

 거듭제곱급수 해법의 개념

– 첨수이동(shift of index)

 

 

 2, 3 3 , 1 5 0 , 2 2 2 , 1 3 0 , 2

2 2 2

2 , 0

2 2

2 2

2 ' '

&

0 4 6 3 5 0 2 4 1 3 0 2

2 2

1

0 1 0

1 2 1

0 1 2

1 1

0 1

1 1

1 0

 



















 













































 











 



 













 



















a a a a a a

a a a a a a

s a a a a s a

x a x

a s a

x a x

ma a x a x x ma

xy y x ma y x a y

s s

s s

s s s s

s s

m m m m

m m m

m m m

m m

m m m m

m m

m – 2 = s m = s

5.1 Power Series Method

 거듭제곱급수 해법의 개념

 

 

      

       

x a x x a

x x x a

a x

x a x a x a x a x a a x a y

a a a

a a a

a a a a

s s s a a a a s s

x a x

a s s x

a x

a m m

x a m m y x ma y x a y

y y

m m m

s s

s s

s s s s

s s m

m m m

m m

m

m m m

m m m

m m

sin

! cos 5

! 3

! 4

! 1 2

!, 5 4 , 5

! 4 3 , 4

! , 3

! 2

, 2 , 1 , 0 1 1 2

2

1 2 1

1 ''

, '

0 '' example

1 0

5 3 1 4

2 0

5 5 4 4 3 3 2 2 1 0 0

1 3 5 0 2 4 1 3 0 2

2 2

0 0

2 0

2

2

2

2 1

1 0







 



   





 



   





















 





 













 

 













































































 















 



 











순환공식, 점화공식 (recursion formula)

School of Mechanical Systems Engineering Engineering Mathematics I

5.2 Theory of the Power Series Method

 거듭제곱급수 해법의 이론

– 기본 개념

• 거듭제곱급수: 무한급수의 형태

• n-번째까지의 부분합(n-th partial sum)

• 나머지(remainder)

• 수렴 및 발산: x = x₁에서 수렴한다







 



















^



2 0 2 0 1 0 0

0 a a x x a x x

x x a

m

m m

     

n

 

ⁿ

n x a a x x a x x a x x

s  ₀ ₁  ₀  ₂  ₀ ²  ₀

 

 _n1



 ₀



^n1 _n2



 ₀



^n2

n x a x x a x x

R

       

1

 

1

0

0 1 1

lims x1 s x a x x s_n x R_n x

m

m m

n n  



^   

 

 수렴값

5.2 Theory of the Power Series Method

 거듭제곱급수 해법의 이론

– 기본 개념

• 수렴

• 세 가지 대표적인 경우

– 경우 1. 무용: x = x₀ 에서만 유일하게 수렴하는 경우 – 경우 2. 보통: 수렴구간 내에서 수렴하는 경우

– 경우 3. 유용: 모든 구간에서 수렴하는 경우 (수렴구간이 무한대)

 

x s

 

x s

 

x n N R_{n 1}  ₁  _{n 1}  for 

R x x ₀ 

중점(midpoint) 수렴반지름

m m m m

m m

a a R a R

lim 1

or 1 lim

1





 







School of Mechanical Systems Engineering Engineering Mathematics I

5.2 Theory of the Power Series Method

 거듭제곱급수 해법의 이론

 

 

0

as

! 1

!

! 1

6 2 1

! 1

1

3 2 0















 



























R

m m m

m a m a a

x x x x m

m m m

m

m 

   

1

as 1 1 1

1 1 1

2 1

1

3 2 0































 









R

a m a a

x x x x x x

m m m

m

m 

 

 











 

 



























R

m m m

m a a a m

x x m e x

m m m

m m x

as 1 0 1

! 1

!

! 1

! 1 2 3 !

1 2

0



  ^{ }

 

2 8 0

8

8 1 8

8 8

1

512 64 1 8

8 4 1

3 3

1 1

9 6 3

0 3























 













 

 







x x

R x R

a a a

x x x x

m m

m m m

m

m m

m m

m



5.2 Theory of the Power Series Method

 거듭제곱급수 해법의 이론

– 거듭제곱급수의 연산

• 항별 미분(termwise differentiation)

• 항별 덧셈(termwise addition)

• 항별 곱셈(termwise multiplication)

• 모든 계수가 0이 됨(vanishing of all coefficients)

– 만일 어떤 거듭제곱급수가 양의 수렴반지름을 갖고, 수렴구간 전체 에서 합이 항등적으로 0이라면, 급수의 모든 계수는 0이다.

       

^

 



 













1

1 0 0

0 '

m

m m m

m

m x x y x ma x x

a x y

          

^

  



























0

0 0

0 0

0

m

m m m m

m m m

m

m x x b x x a b x x

a x g x f

          

^

  

 





























0

0 0 1

1 0 0

0 0

0

m

m m

m m m

m m m

m

m x x b x x ab ab a b x x

a x g x

f 

School of Mechanical Systems Engineering Engineering Mathematics I

5.2 Theory of the Power Series Method

 거듭제곱급수 해법의 이론

– 실수 해석함수(real analytic function)

• 실수함수 f(x)가 수렴반지름 R>0을 갖고, x – x₀의 거듭제곱급수로 나타내어지면, f(x)는 x = x₀에서 해석적이라 한다.

– 거듭제곱급수 해의 존재

 

     

       

 

^0.

and~ at

analytic

~ are ' ~

'' ~ in ~ and~

~,

~,

~, if true is same the Hence . 0 e convergenc

of radius with of

powers in series power a by d represente be

can thus and at

analytic is solution every then , at analytic are in

and If

solution series power of existence

0 0

0 0

0

























x h x x

x r y x q y x p y x h r q p h R

x x x

x

x x x

r y x q y' x p y'' r p, q,

5.3 Legendre’s Equation. Legendre Polynomials

 Legendre의 방정식

Adrien-Marie Legendre

spherical coordinate system



1x²



y''2xy'n



n1



y0

Legendre function 매개변수(parameter), n

 

1 0 ' 1 1

'' 2_{2 2} 



 



 y

x n y n x y x

analytic at x = 0

   







 



















2

2 1

1 0

1 ''

, '

m

m m m

m m m

m m

x a m m y x ma y

x a y

 



1

 

1



2 0

1

0 1

1 2

2

2    















^







 





m m m m

m m m

m

mx x ma x k a x

a m m x n n k



1 ²







1



0



  y nn y dx

x d dx or d

School of Mechanical Systems Engineering Engineering Mathematics I

5.3 Legendre’s Equation. Legendre Polynomials

 Legendre의 방정식

   

   

    

   

   

   

        

  

  



0,1,2,



1 2

1

0 1 2

1 1

2

0 1 2

2 1 2 3 4 2 :

0 1 2

2 3 1 :

0 1 1

2 0 :

0 2

1 1

2

0 2

1 1

0 2

1 1

2

2

2 2

2 4 2

1 1

3 1

0 2

0

0 1

2 0

2

0 1

2 2

2

0 1

1 2

2 2

 







 

























































































































 



























 



























s s a

s s n s a n

a n n s s s a s s

a n n a a a s

x

a n n a a s

x

a n n a s

x

x ka x sa x

a s s x a s s

x ka x ma x

a m m x

a m m

x a k x ma x x a m m x

s s

s s

s s s s

s s s

s s s

s s

m m m m

m m m

m m m

m m

m m m m

m m m

m m

순환공식, 점화공식 (recursion formula)

5.3 Legendre’s Equation. Legendre Polynomials

 Legendre의 방정식

  

    

 

  

    

  

  

    

 

         

    







 





 





 



 

 



































 





 







 













 





 





 



 







 





^





5 1

4 0 3

1 2

0 1

0

5 5 4 4 3 3 2 2 1 0 0

1 3 5

1 3

0 2 4

0 2

2

! 5

4 2 1 3

! 4

3 1 2

! 3

2 1

! 2

1

2 3 4 5

4 2 1 3

4 5

4 3

2 3

2 1

1 2 3 4

3 1 2

3 4

3 2 1 2

1

, 2 , 1 , 0 1

2 1

x n a n n n

x n a n n x n

n a x n

n a x n a a

x a x a x a x a x a a x a x y

n a n n n

n a a n

n a a n

n a n n n

n a a n

n a a n

s s a

s s n s a n

m m m

s s

School of Mechanical Systems Engineering Engineering Mathematics I

5.3 Legendre’s Equation. Legendre Polynomials

 Legendre의 방정식

        

       

     

        

         









 





 



 





 



 

 













         





       



5 3

2

4 2

1

2 1 1 0

5 3

1

4 2

0

! 5

4 2 1 3

! 3

2 1

! 4

3 1 2

! 2 1 1

! 5

4 2 1 3

! 3

2 1

! 4

3 1 2

! 2 1 1

n x n n x n

n x n

x y

n x n n x n

n x n

y

x y a x y a x y

n x n n x n

n x n

a

n x n n x n

n a n x y

5.3 Legendre’s Equation. Legendre Polynomials

 Legendre 다항식, P_n(x)



1 ²

  





1

  

0



  P x nn P x dx

x d dx

d

n n



1 ²







1



0

  y nn y dx

x d dx

d Legendre’s equation

Legendre polynomials

 

x a y

 

x ay

 

x y  _{0 1}  _{1 2}

음이 아닌 정수, n = s

  

  

^s

s a

s s

s n s a n

1 2

1

2  





 

 

  

^





0 m

m mx a x y

   

   

 

:n-th order polynomial 3

2 1 0

3 2 1 0

x P n

x P

n x P

n x

P

n x P

 n























School of Mechanical Systems Engineering Engineering Mathematics I

5.3 Legendre’s Equation. Legendre Polynomials

 Legendre 다항식, P_n(x)

      

:positiveinteger



& 1

! 1 2 5 3 1

! 2

! 2

2    0



 n a

n n n

a_{n n} n 

  

     

    

 

   

   

   

    

    

 

       

  

^{! 2}



^!

! 2

! 2 1 2

! 2

! 1 2

! 2 2

! 2 1

! 1 2

! 2 2 1 2 2 1 2 2

1

! 2

! 2 1 2 2

1 1

2 2 : 1

2

2 1

1 2 1

2 1

2 2

2 2

2 2

m n m n m

m a n

n n a n

n n n n n

n n n n

n n n n n n a n

n n a n

n s

n s s a

n s n

s a s

s a s

s n s a n

n m m n n

n

n n n

n

s s

s s





 



 



 





















 

 

 

 

 









 







 



 







 













     

   

 

integer : 2 or 1 2

! 2

!

! 2

! 2 1 2

0

2

 





 









n M n

m x n m n m

m x n

P

M

m

m n n

m

n Legendre polynomials

5.3 Legendre’s Equation. Legendre Polynomials

 Legendre 다항식, P_n(x)

   

   

   

   

   



x x x x P

x x x

P

x x x P

x x P

x x P

x P

15 70 863 1

3 30 8 35 1

3 2 5 1

1 23 1 1

3 5 5

2 4 4

3 3

2 2

1 0

























 

x P₀

 

^x

P₁

 

x

P₂ P₃

 

x

 

^x

P₄

 

x P₅

School of Mechanical Systems Engineering Engineering Mathematics I

5.4 Frobenius Method

 Frobenius 해법

– 해석적이지 않은 계수를 가지는 특수한 2계 상미분방정식 의 해법을 제공

– 정리 1. Frobenius method

Ferdinand Georg Frobenius

   

0 '

'

'  ₂ y

x x y c x

x y b

   

x^,cx ^:analyticat x⁰ b

     

number complex or real any :

0 ₀

2 2 1 0 0

r

a x

a x a a x x a x x

y ^r

m m m

r     





^





5.4 Frobenius Method

 Frobenius 해법

– 결정방정식(indicial equation)

       

       

       

          

   

       

   

     



1



0

0 1

1 , for

0

1 1

1

1 1

1 ''

1 '

&

0 '

'' 0 '

''

0 0

0 0 0 0

0 0 0 0

2 2 1 0 2

2 1 0

1 0

2 2 1 0 1

0

1 0

2 0

2

1 0

1 0

1

2 2 1 0 0

2 2 1 0

2 2 1 0

2 2













































































































 

































 







 

















c r b r r

a c r b r r a c ra b a r r x

x a x a a x x c x c c

x a r ra x x b x b b x ra r a r r x

x ra r a r r x x a r m r m x y

x a r ra x x a r m x y

x a x a a x x a x x x y

c x c c x c

x b x b b x b

y x c y x xb y x x y

x y c x

x y b

r

r

r r

r

m

r m m

r

m

r m m

r

m m m r















 



School of Mechanical Systems Engineering Engineering Mathematics I

5.4 Frobenius Method

 Frobenius 해법

– 정리 2. 해의 기저

• 경우 1. 두 근의 차가 정수가 아닌 서로 다른 근들

• 경우 2. 이중근

• 경우 3. 두 근의 차가 정수인 서로 다른 근들

   

   

























2 2 1 0 2

2 2 1 0 1

2 1

x A x A A x x y

x a x a a x x y

r r

integer

2 1r  r

2

1 r

r r 

integer

2 1r  r

   

       























0

ln _{1 2} ²

1 2

2 2 1 0 1

x x

A x A x x x y x y

x a x a a x x y

r r





   

     

0 be may

&

0 ln

2 1

2 2 1 0 1

2

2 2 1 0 1

2 1

k r r

x A x A A x x x ky x y

x a x a a x x y

r r































5.4 Frobenius Method

 Frobenius 해법

     

         

   

    

     

   

   

 





  



   



































































 

 

 



 

 

 



 

 

 



















 





 











 











 





 



















2 2 1 0 2

2 1 0 1

2 0

2 0

0 1

0 0

1 0

0

1 0

0

2 0

1 0

2

root double : 0 0 0

0 1

0 3

1 1

1 ''

, '

0 at analytic 1 : 1,

1 3

1 0 ' / 1 / 1 '' 3 1 0

' 1 1 1 '' 3

0 ' 1 3 '' 1 example

x a x a a x

a x a a x x y

r r

a r ra a r r x

x a x

a r m x

a r m

x a r m r m x

a r m r m

x a r m r m y x a r m y x a x y

x x x x x c x x b

x y x y x x

x y x

x y y x x x y x

y y x y x x

r r

m r m m m

r m m m

r m m

m

r m m m

r m m

m

r m m m

r m m m

m m r