PECIAL F UNCTIONS ODE S .S C HAPTER 5. S ERIES S OLUTIONSOF

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C ^HAPTER 5. S ^ERIES S OLUTIONS OF

ODE ^S .

S ^PECIAL F ^UNCTIONS

2018.5 서울대학교 조선해양공학과 서유택

※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다.

Introduction

 In the previous chapters, linear ODEs with constant coefficients (상수계수₎ can be solved by algebraic methods (대수적인 방법), and that their

solutions are elementary functions (초등함수) known from calculus (미적분).

 For ODEs with variable coefficients (변수계수) the situation is more complicated, and their solutions may be nonelementary functions.

 In this chapter, the three main topics are Legendre polynomials, Bessel functions, and hypergeometric functions (초기하함수).

 Legendre’s ODE and Legendre polynomials are obtained by the power series method (거듭제곱급수 또는 멱급수 해법).

 Bessel’s ODE and Bessel functions are obtained by the Frobenius method, an extension of the power series method.





^

^

 

2 2 2

'' ' 0

x y xy  x  y

* Elementary functions (초등함수): 다항 함수, 로그 함수, 지수 함수, 삼각 함수와 이들 함수의 합성 함수들을 총칭

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 Power Series (거듭제곱):

 Coefficients: a₀, a₁ , a₂, …

 Center: x₀

 Power Series in powers of x if x₀ = 0:

 Ex 1 Maclaurin series













0

m m

m

a x x a a x x a x x





      



2

0 1 2

0 m m m

a x a a x a x





   



 

   

   

2 0

2 3

0

2 2 4

0

2 1 3 5

0

1 1 1, geometric series 1

! 1 2! 3!

cos 1 1

2 ! 2! 4!

sin 1

2 1 ! 3! 5!

m m

m x

m

m m

m

m m

m

x x x x

x

x x x

e x

m

x x x

x m

x x x

x x

m













 



     



     

      

      











5.1 Power Series Method (거듭제곱급수해법, 멱급수해법)

 The power series method is the standard method for solving linear ODEs with variable coefficients.

등비급수

5.1 Power Series Method

3 5

*

) sin ,

3! 5!

x x

ex x   x  

x sin x

! 5

! 3

5

3

x

x  x 

! 3 x

x 

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 Idea of the Power Series Method ODE

 We represent p(x) and q(x) by power series in powers of x.

 We assume a solution in the form of a power series with unknown coefficients.

 Series obtained by termwise differentiation

 Insert the series into the ODE.

   

'' ' 0

y  p x y q x y 

2 3

0 1 2 3

0

m m m

y a x a a x a x a x





      

5.1 Power Series Method

1 2

1 2 3

1

' _m ^m 2 3

m

y ma x a a x a x

 





















 ^{ }



2

3 4 2

3 2

) 1

(m a x a a x a x

m

y _m ^m

m

 Ex.2 Solve the following ODE by power series.

Insert the series into the ODE We collect like powers of x.

General solution:

' 0

y  y

2 3

0 1 2 3

0

m m m

y a x a a x a x a x









1

' _m ^m 2 3

m

y ma x a a x a x

 







2 3

2 3

0 0

0 0 0 1 0

2! 3! 2! 3!

a a x x x

y a a x x x a  x  a e

            

 



 



0 0 3 0

1 2

1 0, 2 , 3 , 4 ,

2 2! 3 3! 4 4!

a a a a

a a

a a a a a

       

5.1 Power Series Method

0 )

3 ( ) 2

( )

(a₁ a₀  a₂ a₁ x a₃ a₂ x² 

 , 0 )

3 ( , 0 ) 2

( , 0 )

(a₁ a₀  a₂ a₁  a₃ a₂ x 

Maclaurin series for e^x

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 Ex.3 A Special Legendre Equation

Insert the series into the ODE.

2 3

0 1 2 3

0 m m m

y a x a a x a x a x









1

' _m ^m 2 3

m

y ma x a a x a x

 







5.1 Power Series Method

0 2

2 )

1

(  x² y xy y 















 ^{ }



2

3 4 2

3 2

) 1

(m a x a a x a x

m

y _m ^m

m

General solution:

5 ) 1 3

1 1

( ^{2 4 6}

0

1     

 a x a x x x

y































 

























 

4 4 3

3 2

2 1

0

4 4 3

3 2

2 1

4 4 3

3 2

2 2

4 6 3

5 2

4 3

2

2 2

2 2

2 2

8 6

4 2

2

12 6

2

30 20

12 6

2

x a x

a x

a x

a a

y

x a x

a x

a x

a y

x

x a x

a x

a y

x

x a x

a x

a x

a a

y

 Theory of the Power Series Method

 nth Partial Sum:

 Remainder:

 For example,







 



0

m m

m

a x x a a x x a x x





      



 



 







n

n n

s x  a a x x a x x  a x x

 









n n

n n n

R x a _ x x ^ a _ x x ^ 

5.1 Power Series Method



 





 x x² xⁿ 1

2 3

0 0

2 3 4

1 1

2 3 4 5

2 2

1, ,

1 , ,

1 , ,

s R x x x

s x R x x x

s x x R x x x etc

    

     

      

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 Convergent (수렴): If this sequence converges at x=x₁, say,

→ the series called convergent at x=x₁, s(x₁): convergent value or sum

 Divergent (발산): If this sequence diverges at x=x₁, say,

5.1 Power Series Method

) ( )

(

lim s

x

s x

n















0

0 1

1

) ( )

(

m

m

m

x x

a x

s

n

n x x

s

x x s

x s

s





















 1

, 1

, 1

, 1

2 2

1 0

 



 







n

n n

s x  a a x x a x x  a x x

For example,























 





























2 0

2 0

2

0

2 1

! 1 1

1

! 1 2

!

x x x

m

x x x x

x x m

e x

m

m m

m m

m x

 In the case of convergence,

→ for every n,

 If all s_n(x₁) with n > N lie between s(x₁) ‒ ε and s(x₁) + ε → s(x₁) ≈ s_n(x₁),

 And at x=x₀, s (x₀)=a₀ → the series converges at x₀.

 If there are other values of x for which the series converges besides x₀, there values form an interval → “convergence interval (수렴구간)”

 The series converges for all x in the interval:

The series diverges for all x:

5.1 Power Series Method









0

0 1

1) ( )

(

m

m

m x x

a x

s

) ( )

( )

( x

s x

R x

s 



N n

x s x

s x

R

(

)  (

) 

(

)   for all 

 



 







n

n n

s x  a a x x a x x  a x x

 









n n

n n n

R x a _ x x ^ a _ x x ^ 

R x

x 







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 Convergence Interval (수렴구간). Radius of Convergence (수렴반지름)

Case 1. (useless) The series always converges at the center at x = x₀, s (x₀) = a₀.

Case 2. (usual) If there are further values of x for which the series converges, these values form an interval, called the convergence interval.

5.1 Power Series Method

 



 







n

n n

s x   a a x x   a x x    a x x 

a L x a

x x x a

x x a

m m m m

m

m m

m

  

















1 0

0 1 0

1

lim

) (

) lim (

1 1 1







L L

L

: diverges : inconclusive

R

Radius of convergence of the series (R): | x- x₀ | < R

m m m

m m m

a a x

a x x a

x

1 0

1 0

lim 1 1

lim













   



1

lim

0







 m

m

m

a

x a x

 Convergence Interval. Radius of Convergence

Case 3. (best) The convergence interval may sometimes be infinite, that is, the series converges for all x.

5.1 Power Series Method

0









 0

1 x

x

   R

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5.1 Power Series Method





 

 











 ^{ }



e

m m m

m

x 0,

1 1

! / 1

)!

1 /(

1

! 1 2

!

1 2

0



1 ,

1 1 1 1

1

1 _{2 1}

0















 

 



x _m

m m

m 

0 ,

! 1 )!

1 2 (

1

! ^{2 1}

0









 









 ^





m m

m 

1

1 1 / ( 1)! 1

lim lim lim 0, for all x, the series converges

1 / ! 1

m m

m m m m

m

a x m

x x R

a x m m

 

  

      



1

1 1

lim lim only for 1, the series converges 1

m m

m m m

m

a x

x x x R 1

a x





       

1

1 ( 1)!

lim lim lim ( 1) , for any x, the series diverge 0

!

m m

m m m m

m

a x m

x x m R

a x m





  

       

For example,

5.1 Power Series Method

 Example

m m

m m

k x

2 0





 ) (

m m

m

x²

0 3







 





Seoul

 Theorem 1 Existence of Power Series Solutions

If p, q, and r in yʺ(x)+p(x)yʹ+q(x)y = r(x) are analytic at x = x₀, (power series representations)

then every solution is analytic and can thus be represented by a power series in powers of x- x₀ with radius of convergence R > 0.

5.1 Power Series Method

* 함수f가 한 점 x₀에서 해석적이라는 것은 그 점 근방에서의 테일러 급수가 수렴하는 것과 같은 의미

 Legendre’s equation:

Substituting and its derivatives

The first expression as two separate series.

Set m‒2 = s in the first series

and simply write s instead of m in the other three series.





^

^

0 m m m

y a x











 ^{ }

^{ }

2 1 0

1 1 _m ^m 2 _m ^m 1 _m ^m 0

m m m

x m m - a x x ma x n n a x

  

 

  









5.2 Legendre’s Equation. Legendre Polynomials P

(x)

0 )

1 ( 2

) 1 (

) 1 (

0 1

2 2

2      



  



















m

m m m

m m m

m m m

m

mx m m a x ma x n n a x

a m

m

0 )

1 ( 2

) 1 ( )

1 )(

2 (

0 1

2 0

2      





  















 

s

s s s

s s s

s s s

s

s x s s a x sa x n n a x

a s

s

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 Recurrence relation (점화관계):

  

    

2

1 0 1

2 1

s s

n s n s

a a s , ,

s s



  

  

 

5.2 Legendre’s Equation. Legendre Polynomials P

(x)

 The sum of the coefficients of each power of x on the left must be zero.

0 )

1 ( 2

) 1 ( )

1 )(

2 (

0 1

2 0

2      





  















 

s

s s s

s s s

s s s

s

s x s s a x sa x n n a x

a s

s

0 2

0 0 0

2 0

! 2

) 1 0 (

) 1 ( 1

2

: n n a

a x

a n

n x a

x 

















1 3

1 3

1

! 3

) 2 )(

1 0 (

)]

1 ( 2 [ 2

3

: n n a

a a

n n a

x  





















2 2

( )( 1)

: ( 2)( 1) [ ( 1) 2 ( 1)] 0

( 2)( 1)

s

s s s s

n s n s

x s s a s s s n n a a a

s s

 

  

           

 

 

  

    

  

  

    

2 0 3 1

4 2 5 3

0 1

1 1 2

2! 3!

2 3 3 4

4 3 5 4

2 1 3 3 1 2 4

4! 5!

n n n n

a a a a

n n n n

a a a a

n n n n n n n n

a a

  

   

   

   

 

      

 

        

         

2 4

1

3 5

2

1 2 1 3

1 2! 4!

1 2 3 1 2 4

3! 5!

n n n n n n

y x x x

n n n n n n

y x x x x

   

    

     

    

 

 

 

y x  a y x a y x

5.2 Legendre’s Equation. Legendre Polynomials P

(x)

 We express successively except for a₀ and a₁.

 General solution:

0

m m m

y a x









 y₁ and y₂ are independent

 x=±1 → no longer analytic





^

^

  

    

2

1 0 1

2 1

s s

n s n s

a a s , ,

s s



  

  

 

Seoul

 Legendre Polynomials

The reduction of power series ( , m→ ∞) to polynomials (m is finite) is a great advantage

because then we have solutions for all x, without convergence restrictions.

5.2 Legendre’s Equation. Legendre Polynomials

0 m m m

y a x









        

         

2 4

1

3 5

2

1 2 1 3

1 2! 4!

1 2 3 1 2 4

3! 5!

n n n n n n

y x x x

n n n n n n

y x x x x

   

    

     

    

  

    

2

1 0 1

2 1

s s

n s n s

a a s , ,

s s



  

  

 

2 0, 4 0, 6 0,...

n n n

a _  a _  a _ 



   







 ₂ ² ₄ ⁴ _2 ^2 _2 ^2

1 1 a x a x a_n xⁿ a_nxⁿ a_n xⁿ

y



   





 ₁ ¹ ₃ ³ _1 ^1 _1 ^1 _3 ^3

2

n n n

n n

n x a x a x

a x

a x a y If s=n

If n is odd If n is even

Polynomials (finite terms)

Polynomials (finite terms)

“Legendre polynomials”





^

^

 Legendre Polynomials

The reduction of power series ( , m → ∞) to polynomials (m is finite) is a great advantage

because then we have solutions for all x, without convergence restrictions.

5.2 Legendre’s Equation. Legendre Polynomials

0 m m m

y a x









        

         

2 4

1

3 5

2

1 2 1 3

1 2! 4!

1 2 3 1 2 4

3! 5!

n n n n n n

y x x x

n n n n n n

y x x x x

   

    

     

    

  

    

2

1 0 1

2 1

s s

n s n s

a a s , ,

s s



  

  

 

2 0, 4 0, 6 0,...

n n n

a _  a _  a _ 



   







 ₂ ² ₄ ⁴ _2 ^2 _2 ^2

1 1 a x a x a_n xⁿ a_nxⁿ a_n xⁿ

y



   





 ₁ ¹ ₃ ³ _1 ^1 _1 ^1 _3 ^3

2

n n n

n n

n x a x a x

a x

a x a y If s=n

If n is odd If n is even

Polynomials (finite terms)

Polynomials (finite terms)

“Legendre polynomials”





^

^

Seoul

 Legendre Polynomials

We choose the coefficient a_n of the highest power xⁿ as

We solved for a_s in terms of a_s+2

  

  

 

2 1

2

s 1 s

s s

a a s n

n s n s ^

 

   

  

5.2 Legendre’s Equation. Legendre Polynomials

  

    

2

1 0 1

2 1

s s

n s n s

a a s , ,

s s



  

  

 

n p_n( 1) 1 for every

)!

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   







 



 

 

 

 

n n

n n

n

n n

n n

n n n

n n

n a n

n n

a_n n _{n n n}

2

(2 2)!

2 ( 1)!( 2)!

n n

a n

n n



  

 

The choice of a_n makes

 

 

 

1 a positive integer 2 !

1 3 5 2 1 2 ! 0

!

n n

n n

a n

n n

n



    

 

for n = 0

for n ≠ 0, n is a positive integer

2 s  n

 Legendre Polynomials

     

   

0

2 2 ! 1

1 or

2 ! ! 2 ! 2 2

M m n m

n n

m

n m n n

P x x M

m n m n m





   





   

   

2

2 2 !

2 0, 1

2 ! ! 2 !

m

n m n

n m

n m a

m n m n m



    

 

5.2 Legendre’s Equation. Legendre Polynomials

)!

2 ( )!

1 ( 2

)!

2 2 (

2  

 

 n n

a_{n n} n

)!

4 ( )!

2 (

! 2 2

)!

4 2

( )

3 2 ( 4

) 3 )(

2 (

2

4  

 

 



 

 _

 n n

a n n

n

a_n n _{n n}

When n-2m ≥ 0

  

  

 

2 1

2

s 1 s

s s

a a s n

n s n s ^

 

   

  

whichever is an integer

4 s  n

2 2

(2 )! (2 2)!

2 ( !) 2 1!( 1)!( 2)!

n n

n n

n n

x x

n n n

 

  

 

Seoul

 Examples of the Legendre Polynomials

5.2 Legendre’s Equation. Legendre Polynomials

     

   

0

2 2 ! 1

1 or

2 ! ! 2 ! 2 2

M m n m

n n

m

n m n n

P x x M

m n m n m





   









^

^

 

0 0 2

0! 1

2 (0!)

P x  x  1

 

2!

2 (1!)

P x  x  x

2 2

(2 )! (2 2)!

2 ( !) 2 1!( 1)!( 2)!

n n

n n

n n

x x

n n n



  

 

Q: Derive P

(x) and prove it satisfies the ODE.

 Indicial Equation (결정방정식), Indicating the Form of Solutions

 We expand b(x) and c(x) in power series

 We differentiate term by term

 The equation corresponding to the power x^r is .

 By assumption a₀ ≠ 0, Indicial Equation:

   

2 '' ' 0

x y xb x y c x y 

   





 



 

 

 



r r r

x r r  a   b b x x ra   c c x x a a x 

 1 

0

r r   b r  c 

   

'' b x ' c x2 0

y y y

x x

  

  



0 0

r m m r r

m m

m m

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

 



 







 1 

0

 





 

 

5.3 Extended Power Series Method: Frobenius Method

Multiply x²

] )

1 ( [

) (

)

( ₁

0

0 1

1    



 









 x ra r a x

x a r m x

y

m

r r

m m

] )

1 ( )

1 ( [ )

1 )(

( )

( ₁

0

0 2

2     







 









 x r r a r ra x

x a r

m r m x

y

m

r r

m m

doesn’t change!

Seoul

 Theorem 2 Frobenius Method. Basis of Solutions. Three Cases

Suppose that the ODE satisfies the assumptions in Theorem 1. Let r₁ and r₂ be the roots of the indicial equation. Then we have the following three cases.

Case 1. Distinct Roots Not Differing by an Integer. A basis is

Case 2. Double Root (r₁ = r₂ =r). A basis is

Case 3. Roots Differing by an Integer. A basis is

where the roots are so denoted that r₁ - r₂ > 0.

 



 ^{ }





1 ^r 0 1 2 and 2 ^r 0 1 2

y x  x a a x a x  y x  x A  A x A x 

  

 ^{   } 



1 ^r 0 1 2 and 2 1 ln ^r 1 2

y x  x a a x a x  y x  y x x x A xA x 

 



 ^{   }





1 ^r 0 1 2 and 2 1 ln ^r 0 1 2

y x  x a a x a x  y x ky x x x A  A x A x 

5.3 Extended Power Series Method: Frobenius Method

(두 근의 차가 정수가 아닌 서로 다른 근들)

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