The Chemistry Maths Book, Second Edition

340 Chapter 12Second-order differential equations. Constant coefficients

EXAMPLE 12.2The general solution of the equation in Example 12.1 is

y(x) 1 = 1 c

1

e

2 x

1 + 1 c

2

e

3 x

Particular solutions are obtained by assigning particular values toc

1

andc

2

0 Exercises 4–6

12.3 The general solution

Let one solution of equation (12.3),

(12.3)

be

y 1 = 1 e

λx

(12.6)

Then

and substitution into (12.3) gives

λ

2

e

λx

1 + 1 aλe

λx

1 + 1 be

λx

1 = 10

or

e

λx

(λ

2

1 + 1 aλ 1 + 1 b) 1 = 10

The function (12.6) is therefore a solution of equation (12.3) if

λ

2

1 + 1 aλ 1 + 1 b 1 = 10 (12.7)

This quadratic equation is called the characteristic equationor auxiliary equationof

the differential equation.

2

The possible values of λare the roots of the quadratic:

(12.8)

λλ

1

2

2

2

1

2

4

1

2

=−+ − 4

( )

=−− −

( )

aa b, aa b

d

dx

ee

d

dx

ee

λλxx λ λx x

=,λλ=

2

2

2

dy

dx

a

dy

dx

by

2

2

++= 0

2

In a letter to Johann Bernoulli in 1739 Euler described his discovery of the solution of linear equations with

constant coefficients by means of the characteristic equation: ‘after treating the problem in many ways, I happened

on my solution entirely unexpectedly; before that I had no suspicion that the solution of algebraic equations had

so much importance in this matter’.