Page 90 - Schaum's Outline of Differential Equations

CHAPTER 8

Linear Differential

Equations: Theory

of Solutions

LINEAR DIFFERENTIAL EQUATIONS
An nth-order linear differentia! equation has the form

where g(x) and the coefficients />, (.v) ( / = (). i. 2 ..... n) depend solelj on the variable x. In other words. the;
do not depend on y or on any derivative of y,
If g(x) =0. then hq. (8.1) is homogeneous', if not, (8.1) is nonhomogeneous. A linear differential equation
has constant coefficients if all the coefficients bj(x) in (8.1) are constants: if one or more of these coefficients is
not constant, (8.1) has variable coefficients.
Theorem 8.1. Consider the initial-; aluc problem given h\ the linear differential equation (8.1) and the n initial
conditions

If £(A) and h t(x) (j = 0. 1. 2. ... , n) arc continuous in some interval 3 containing A () and if
b,,(x)^Q in ,'/. then the initial-value problem given hy (8.1) and (8.2) has a unique (only one)
solution defined throughout ,<7.
When the conditions on /^(.v) in 'I'heorem 8.1 hold, we can divide Hq. <<S'./) by b,,(x) to get

where Oj(x) = bj<x)/b,,(x) ( / = 0. I ..... n - 1 ) and <j>(x) = g(x)/b,,(x).
Let US define ihe dilTerential operator L(y) by

where a,<jr) (i = 0, I. 2 ..... n— I) is continuous on some interval of interest. Then (8.3) can be rewritten as

and. in particular, a linear homogeneous differential equation can he expressed as

73
Copyright © 2006, 1994, 1973 by The McGraw-Hill Companies, Inc. Click here for terms of use.