Chapter 64

The solution of differential

equations using Laplace

transforms

64.1 Introduction

An alternative method of solving differential equations
to that used in Chapters 46 to 51 is possible by using
Laplace transforms.

64.2 Procedure to solve differential

equations by using Laplace

transforms

(i) Take the Laplace transform of both sides of the

differential equation by applying the formulae

for the Laplace transforms of derivatives (i.e.

equations (3) and (4) of Chapter 62) and, where

necessary, using a list of standard Laplace trans-

forms, such as Tables 61.1 and 62.1 on pages 584

and 587.

(ii) Put in the given initial conditions, i.e. y( 0 )

andy′( 0 ).

(iii) Rearrange the equation to makeL{y}the subject.

(iv) Determineyby using, where necessary, partial

fractions, and taking the inverse of each term by

using Table 63.1 on page 593.

64.3 Worked problemson solving

differential equations using

Laplace transforms

Problem 1. Use Laplace transforms to solve the

differential equation

2

d^2 y

dx^2

+ 5

dy

dx

− 3 y= 0 ,given that whenx= 0 ,

y=4and

dy

dx

=9.

This is the same problem as Problem 1 of Chapter 50,

page 478 and a comparison of methods can be made.

Using the above procedure:

(i) 2L

{

d^2 y

dx^2

}

+ 5 L

{

dy

dx

}

− 3 L{y}=L{ 0 }

2[s^2 L{y}−sy( 0 )−y′( 0 )]+5[sL{y}

−y( 0 )]− 3 L{y}= 0 ,

from equations (3) and (4) of Chapter 62.