Differential Equations and Linear Algebra, 4th Edition

5 Higher-Order Linear Differential Equations

5.1 Introduction: Second-Order Linear Equations

In Chapters 1 and 2 we investigated first-order differential equations. We now turn to equations of higher order $n\geqq 2\text{,}$ beginning in this chapter with equations that are linear. The general theory of linear differential equations parallels the second-order case ($n=2$), which we outline in this initial section.

Recall that a second-order differential equation in the (unknown) function y(x) is one of the form

$$G(x,y,y\text{′},y″)=0.$$ (1)

This differential equation is said to be linear provided that G is linear in the dependent variable y and its derivatives $y\text{′}$ and $y″\text{.}$ Thus a linear second-order equation takes (or can be written in) the form

$$A(x)y″+B(x)y\text{′}+C(x)y$$