# 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

This differential equation is said to be **linear** provided that `G` is linear in the dependent variable `y` and its derivatives $y\text{\u2032}$ and $y\u2033\text{.}$ Thus a linear second-order equation takes (or can be written in) the form

Get *Differential Equations and Linear Algebra, 4th Edition* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.