Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

coefficient of the highest powerzN; such a power now exists because of our

assumed form of solution.

By assuming a polynomial solution find the values ofλin (16.34) for which such a solution

exists.

We assume a polynomial solution to (16.34) of the formy=

∑N

n=0anz

n. Substituting this

form into (16.34) we find

∑N

n=0

[

n(n−1)anzn−^2 − 2 znanzn−^1 +λanzn

]

=0.

Now, instead of starting with the lowest power ofz, we start with the highest. Thus,
demanding that the coefficient ofzNvanishes, we require− 2 N+λ=0,i.e.λ=2N,aswe
found in the previous example. By demanding that the coefficient of a general power ofz
is zero, the same recurrence relation as above may be derived and the solutions found.

16.6 Exercises

16.1 Find two power series solutions aboutz= 0 of the differential equation

(1−z^2 )y′′− 3 zy′+λy=0.
Deduce that the value ofλfor which the corresponding power series becomes an
Nth-degree polynomialUN(z)isN(N+ 2). ConstructU 2 (z)andU 3 (z).
16.2 Find solutions, as power series inz, of the equation

4 zy′′+2(1−z)y′−y=0.

Identify one of the solutions and verify it by direct substitution.
16.3 Find power series solutions inzof the differential equation

zy′′− 2 y′+9z^5 y=0.

Identify closed forms for the two series, calculate their Wronskian, and verify
that they are linearly independent. Compare the Wronskian with that calculated
from the differential equation.
16.4 Change the independent variable in the equation

d^2 f

dz^2

+2(z−a)

df

dz

+4f=0 (∗)

fromztox=z−α, and find two independent series solutions, expanded about

x= 0, of the resulting equation. Deduce that the general solution of (∗)is

f(z, α)=A(z−α)e−(z−α)

2

+B

∑∞

m=0

(−4)mm!

(2m)!

(z−α)^2 m,

withAandBarbitrary constants.
16.5 Investigate solutions of Legendre’s equation at one of its singular points as
follows.

(a) Verify thatz= 1 is a regular singular point of Legendre’s equation and that

the indicial equation for a series solution in powers of (z−1) has roots 0

and 3.

(b) Obtain the corresponding recurrence relation and show thatσ= 0 does not

give a valid series solution.