Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

232

Definition The point x 0 is a regular singular point , of a differential equation if this equation can be represented around the point x 0 in the form ( x − x 0 ) n y ( n ) + b 1 ( x )( x − x 0 ) ( n − 1 ) y ( n − 1 ) + ··· + b n ( x ) y = 0 , (5.33) where b i ( x ) , i = 1 , 2 ,..., n , are analytical functions at point x 0 .

As we will tackle later only second-order differential equations, let us consider the equation

2 y ′′ + b ( x )( x − x

0 ) y ′ + c ( x ) y = 0 .

L ( y ) ≡ ( x − x 0 )

(5.34)

Without losing generality, and for the sake of simpler writing, we will assume that x 0 = 0 4 , and thus obtain L ( y ) ≡ x 2 y ′′ + b ( x ) xy ′ + c ( x ) y = 0 . (5.35) We assume that b ( x ) and c ( x ) are analytical functions at point x , that is, that there exists a number R > 0 such that they can be represented by power series

∞ ∑ k = 0

∞ ∑ k = 0

k , c ( x )=

k ,

b ( x )=

b k x

c k x

(5.36)

that converge in the interval | x | < R . We will seek the solution in the form of a so called generalized power series

∞ ∑ k = 0

λ

k , a

y ( x )= x

a k x

0̸ = 0 , x > 0 .

(5.37)

Theorem 14 (Frobenius 5 method) Every differential equation of the form

b ( x ) x

c ( x ) x 2

y ′′ +

y ′ +

y = 0 ,

(5.38)

where functions b ( x ) and c ( x ) are analytical at point x = 0, has a solution in the form

∞ ∑ k = 0

λ

k , a

y ( x )= x

a k x

0̸ = 0 ,

(5.39)

where λ can be any number (real or complex). λ is chosen so that a 0̸ = 0.

Substituting the assumed solution into the initial equation, for a 0̸ = 0 and k = 0we obtain the equation for determining λ λ ( λ − 1 )+ b ( 0 ) λ + c ( 0 )= 0 . (5.40) 4 We could introduce a new variable x = x − x 0 and thus formally obtain the same equation, as for x 0 = 0. 5 Georg Frobenius (1849-1917), German mathematician, who contributed significantly to matrix theory and group theory.