14

First-order ordinary differential

equations

Differential equations are the group of equations that contain derivatives. Chap-

ters 14–21 discuss a variety of differential equations, starting in this chapter and

the next with those ordinary differential equations (ODEs) that have closed-form

solutions. As its name suggests, an ODE contains only ordinary derivatives (no

partial derivatives) and describes the relationship between these derivatives of

thedependent variable, usually calledy, with respect to theindependent variable,

usually calledx. The solution to such an ODE is therefore a function ofxand

is writteny(x). For an ODE to have a closed-form solution, it must be possible

to expressy(x) in terms of the standard elementary functions such as expx,lnx,

sinxetc. The solutions of some differential equations cannot, however, be written

in closed form, but only as an infinite series; these are discussed in chapter 16.

Ordinary differential equations may be separated conveniently into differ-

ent categories according to their general characteristics. The primary grouping

adopted here is by theorderof the equation. The order of an ODE is simply the

order of the highest derivative it contains. Thus equations containingdy/dx, but

no higher derivatives, are called first order, those containingd^2 y/dx^2 are called

second order and so on. In this chapter we consider first-order equations, and in

the next, second- and higher-order equations.

Ordinary differential equations may be classified further according todegree.

The degree of an ODE is the power to which the highest-order derivative is

raised, after the equation has been rationalised to contain only integer powers of

derivatives. Hence the ODE

d^3 y

dx^3

+x

(

dy

dx

) 3 / 2

+x^2 y=0,

is of third order and second degree, since after rationalisation it contains the term

(d^3 y/dx^3 )^2.

Thegeneral solutionto an ODE is the most general functiony(x) that satisfies

the equation; it will containconstants of integrationwhich may be determined by