Page 279 - Schaum's Outline of Differential Equations

CHAPTER 27

Power Series Solutions

of Linear Differential

Equations with

Variable Coefficients

SECOND-ORDER EQUATIONS
A second-order linear differential equation

has variable coefficients when b 2(x), b^x), and b Q(x) are not all constants or constant multiples of one another.
If b 2(x) is not zero in a given interval, then we can divide by it and rewrite Eq. (27.1) as

where P(x) = b 1(x)lb 2(x), Q(x) = b 0(x)lb 2(x), and (f)(x) = g(x)lb 2(x). In this chapter and the next, we describe
procedures for solving many equations in the form of (27.1) or (27.2). These procedures can be generalized
in a straightforward manner to solve higher-order linear differential equations with variable coefficients.

ANALYTIC FUNCTIONS AND ORDINARY POINTS

A function/^) is analytic at x 0 if its Taylor series about x 0,

converges tof(x) in some neighborhood of x 0.
Polynomials, sin x, cos x, and e* are analytic everywhere; so too are sums, differences, and products of these
functions. Quotients of any two of these functions are analytic at all points where the denominator is not zero.
The point x 0 is an ordinary point of the differential equation (27.2) if both P(x) and Q(x) are analytic at x 0.
If either of these functions is not analytic at x 0, then x 0 is a singular point of (27.2).

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