The Chemistry Maths Book, Second Edition

11 First-order differential equations

11.1 Concepts

A differential equation is an equation that contains derivatives. For example, if yis a

function of xthen an equation that contains yand xonly is an ordinary equation, but

an equation that also contains one or more ofdy 2 dx,d

2

y 2 dx

2

, and higher derivatives

is a differential equation. Examples of differential equations, with representative uses

in the physical sciences, are

1. first-order rate process


2. second-order rate process


3. body falling under the influence of gravity


4. classical harmonic oscillator


5. wave equation for the harmonic oscillator

The five examples describe physical processes.

1

Equations 1 and 2 are used to describe

(to model) some simple rate processes in physics and chemistry, and also in biology,

engineering, economics, and the social sciences. Equations 3 , 4 , and 5 are equations

of motion. In all of these the unknown function is a function of one variable only,

x(t)in 1 , 2 , and 4 ,h(t)in 3 ,ψ(x)in 5. The derivatives are ordinary derivatives and

the equations are ordinary differential equations. Equations containing partial

derivatives are discussed in Chapter 14.

There exists a system of classification of the possible types of differential equations,

but for our purposes we consider only the order. The order of a differential equation

is the order of the highest derivative in the equation; equations 1 and 2 are first-order

equations because they contain a first derivative only, equations 3 to 5 are second-

order equations. Almost all the important differential equations in the physical

sciences are first order (such as elementary rate processes) or second order (equations

of motion such as Newton’s second law, the Maxwell equations in electromagnetism,

and the Schrödinger equation).

−

+=



22

2

2

2

1

m 2

d

dx

kx E

ψ

ψψ

dx

dt

x

2

2

2

=−ω

dh

dt

g

2

2

=−

dx

dt

=− −ka x b x()()

dx

dt

=kx

1

Both Newton and Leibniz recognized that physical problems can be formulated in terms of differential

equations, and the solution of physical problems provided much of the motivation for the further development of

the calculus in the 18th century.