Chapter 65

The solution of simultaneous

differential equations using

Laplace transforms

65.1 Introduction

It is sometimes necessary to solve simultaneous differ-
ential equations. An example occurs when two electrical
circuits are coupled magnetically where the equations
relating the two currentsi 1 andi 2 are typically:

L 1

di 1

dt

+M

di 2

dt

+R 1 i 1 =E 1

L 2

di 2

dt

+M

di 1

dt

+R 2 i 2 = 0

whereLrepresents inductance,Rresistance,Mmutual
inductance andE 1 the p.d. applied to one of the circuits.

65.2 Procedure to solve simultaneous

differential equations using

Laplace transforms

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

simultaneous equation by applying the formu-

lae for the Laplace transforms of derivatives (i.e.

equations(3) and (4)of Chapter62, page 589)and

using a list of standard Laplace transforms, as in

Table 61.1, page 584 and Table 62.1, page 587.

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

y′( 0 ).

(iii) Solve the simultaneous equations forL{y}and

L{x}by the normal algebraic method.

(iv) Determineyandxby using, where necessary,

partial fractions, and taking the inverse of each

term.

65.3 Worked problems on solving

simultaneous differential

equations by using Laplace

transforms

Problem 1. Solve the following pair of

simultaneous differential equations

dy

dt

+x= 1

dx

dt

−y+4et= 0

given that att=0,x=0andy=0.

Using the above procedure:

(i) L

{

dy

dt

}

+L{x}=L{ 1 } (1)