Page 293 - Schaum's Outline of Differential Equations

276 SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT [CHAP. 28

and

are substituted into Eq. (28.1). Terms with like powers of x are collected together and set equal to zero. When
this is done for x" the resulting equation is a recurrence formula. A quadratic equation in A,, called the indicial
equation, arises when the coefficient of x° is set to zero and a 0 is left arbitrary.
The two roots of the indicial equation can be real or complex. If complex they will occur in a conjugate
pair and the complex solutions that they produce can be combined (by using Euler's relations and the identity
xa± ib _ xa g ± ib in ^ f orm reaj solutions. In this book we shall, for simplicity, suppose that both roots of the
to
indicial equation are real. Then, if A, is taken as the larger indicial root, A = A x > A 2, the method of Frobenius
always yields a solution

to Eq. (28.1). [We have written a n(k]) to indicate the coefficients produced by the method when A = A x.]
If P(x) and Q(x) are quotients of polynomials, it is usually easier first to multiply (28.1) by their lowest
common denominator and then to apply the method of Frobenius to the resulting equation.

GENERAL SOLUTION

The method of Frobenius always yields one solution to (28.1) of the form (28.5). The general solution (see
Theorem 8.2) has the form y = c^y^x) + C 2y 2(x) where c 1 and c 2 are arbitrary constants and y 2(x) is a second
solution of (28.1) that is linearly independent from yi(x). The method for obtaining this second solution depends
on the relationship between the two roots of the indicial equation.
Case 1. If A! - A 2 is not an integer, then

m
where y 2(x) is obtained in an identical manner as y\(x) by the method of Frobenius, using A^ place of A x.
Case 2. If A x = A 2, then

To generate this solution, keep the recurrence formula in terms of A and use it to find the coefficients
a n (n > 1) in terms of both A and a 0, where the coefficient a 0 remains arbitrary. Substitute these a n into
Eq. (28.2) to obtain a function y(k, x) which depends on the variables A and x. Then

Case 3. If A x - A 2 = N, a positive integer, then

To generate this solution, first try the method of Frobenius. with A 2. If it yields a second solution, then
this solution is y 2(x), having the form of (28.9) with d^ = 0. Otherwise, proceed as in Case 2 to generate
y(k, x), whence