The Chemistry Maths Book, Second Edition

318 Chapter 11First-order differential equations

All ordinary differential equations can be solved by the numerical methods discussed

in Chapter 20. There are however several important standard types whose solutions

can be expressed in terms of elementary functions and that frequently occur in

mathematical models of physical systems. It is these standard types that are discussed

in this and the following chapters.

The general first-order differential equation has the form

(11.8)

whereF(x, y)is a function of xand y, and yis a function of x. Such a differential

equation together with an initial condition,y(x

0

) 1 = 1 y

0

(y 1 = 1 y

0

whenx 1 = 1 x

0

), is called an

initial value problem. We discuss here two important special types of equations that

can be solved by elementary methods; separable equations and linear equations.

0 Exercises 10–12

11.3 Separable equations

Equation (11.8) can be solved by straightforward integration if it has, or if it can be

reduced to, the form

(11.9)

This equation can be written in the differential form

2

g(y) dy 1 = 1 f(x) dx (11.10)

with all the terms involving yon one side of the equal sign and all the terms involving

xon the other. Such an equation is called a separable differential equationand is

solved by (indefinite) integration of each side with respect to the relevant variable:

(11.11)

with the arbitrary constant included on one side. More formally, writing (11.9) as

(11.12)

integration with respect to xgives

ZZgy (11.13)

dy

dx

() dx=+f x dx c()

gy

dy

dx

() =fx()

ZZg y dy() =+f x dx c()

dy

dx

fx

gy

=

()

()

dy

dx

=,Fxy()

2

Much of Leibniz’s formulation of the calculus involved equations containing differentials (differential

equations). The method of separation of variables and the method of reducing a homogeneous equation to

separable form discussed in this section were invented by Leibniz in 1691.