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 n2, 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,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 and y. Thus a linear second-order equation takes (or can be written in) the form

A(x)y+B(x)y+C(x)y

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