15

Higher-order ordinary differential

equations

Following on from the discussion of first-order ordinary differential equations

(ODEs) given in the previous chapter, we now examine equations of second and

higher order. Since a brief outline of the general properties of ODEs and their

solutions was given at the beginning of the previous chapter, we will not repeat

it here. Instead, we will begin with a discussion of various types of higher-order

equation. This chapter is divided into three main parts. We first discuss linear

equations with constant coefficients and then investigate linear equations with

variable coefficients. Finally, we discuss a few methods that may be of use in

solving general linear or non-linear ODEs. Let us start by considering some

general points relating toalllinear ODEs.

Linear equations are of paramount importance in the description of physical

processes. Moreover, it is an empirical fact that, when put into mathematical

form, many natural processes appear as higher-order linear ODEs, most often

as second-order equations. Although we could restrict our attention to these

second-order equations, the generalisation tonth-order equations requires little

extra work, and so we will consider this more general case.

A linear ODE of general ordernhas the form

an(x)

dny

dxn

+an− 1 (x)

dn−^1 y

dxn−^1

+···+a 1 (x)

dy

dx

+a 0 (x)y=f(x). (15.1)

Iff(x) = 0 then the equation is calledhomogeneous; otherwise it isinhomogeneous.

The first-order linear equation studied in subsection 14.2.4 is a special case of

(15.1). As discussed at the beginning of the previous chapter, the general solution

to (15.1) will containnarbitrary constants, which may be determined ifnboundary

conditions are also provided.

In order to solve any equation of the form (15.1), we must first find the

general solution of thecomplementary equation, i.e. the equation formed by setting