Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

228

Theorem13 If the functions p , q and r , in differential equation

y ′′ + p ( x ) y ′ + q ( x ) y = r ( x )

(5.20)

are analytical at point x = x 0 , then each solution of the equation (5.20) is analytical at point x = x 0 , and can be represented by a power series of ( x − x 0 ) , with a convergence radius R > 0.

R Note. When applying this theorem, it is important to write the equation in the form (5.20), so that the coefficient next to the highest derivative is equal to 1.

Finally, let us note that this method is of significant practical importance due to the possibility of calculating numerical values for a series.

5.3 Legendre: equation, function, polynomial

Definition Differential equation of the form

( 1 − x 2 ) y ′′ − 2 xy ′ + k ( k + 1 ) y = 0 ,

(5.21)

where k is a known real number, is known in the literature as Legendre equation . Solutions of the equation (5.21) are called Legendre functions .

This equation appears in numerous problems of mathematical physics, as well as in solving partial differential equations. According to the previous note, the coefficient next to y ′′ should be equal to 1, and thus dividing by ( 1 − x 2 ) we obtain y ′′ − 2 x 1 − x 2 y ′ + k ( k + 1 ) 1 − x 2 y = 0 . (5.22) As the conditions of Theorem 13 are fulfilled p = − 2 x 1 − x 2 , q = k ( k + 1 ) 1 − x 2 , r = 0 , that is, the corresponding coefficients ( p , q , r ) are analytical functions for x = 0 , the solution can be represented by the power series

∞ ∑ i = 0

i .

y =

a i x

(5.23)

Coefficients a i are determined from the condition that this solution satisfies the initial equation for each x . By substituting the assumed solution into the initial equation, we obtain 1 − x 2 ∞ ∑ i = 2 i ( i − 1 ) a i x i − 2 − 2 x ∞ ∑ i = 1 ia i x i − 1 + k ( k + 1 ) ∞ ∑ i = 0 a i x i = 0 , (5.24)