Objectives
In this chapter you will find:- definitions and terminology for differential equations
- initial and boundary conditions
- direct integration and separation of variables of first order equations
- linear equations and integrating factors
- second order linear homogeneous equations – auxiliary equation
- second order linear inhomogeneous equations – complementary function and
particular integral
Motivation
You may need the material of this chapter for:- modelling the motion of particles in mechanics
- modelling the time behaviour of electrical and electronic circuits
- the study of fluid flow
- modelling of chemical reactions
- modelling economic and financial systems and manufacturing processes
15.1 Introduction
Differential equations are often introduced by talking about rates at which radioactive
substances decay. All very well, but how many of us have a nice handy sample of pluto-
nium to play with? Now, bacterial growth – plenty of that found on mouldy bread in the
average student kitchen. If you can contain your queasiness long enough to take a detached
view of it, that grey, dusty, inedible square of once-bread can in fact provide a ready made
mini-laboratory for introducing differential equations.
If you keep a record of the growth of the mould with sufficient accuracy (or, better, let
somebody else do it) you will find that the rate at which the bacteria multiply at a given
time is roughly proportional (14
➤
) to the number present at that time. Mathematically:
ifnis the number of bacteria present at timet,thennis a function oft, which we write
n=n(t), and it satisfies
dn
dt∝nor
dn
dt
=kn ( 15. 1 )wherekis some constant. This is called adifferential equation(DE) for thedependent
variablenin terms of theindependent variablet. It’s a little bit more complicated than
it need be, so let’s tidy it up.