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.