Chapter 65
The solution of simultaneous
differential equations using
Laplace transforms
65.1 Introduction
It is sometimes necessary to solve simultaneous differential equations. An example occurs when two electrical
circuits are coupled magnetically where the equations
relating the two currents i 1 and i 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
where L represents inductance, R resistance, M mutual
inductance and E 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 formulae for the Laplace transforms of derivatives (i.e.
equations (3) and (4) of Chapter 62, 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 for L{y} and
L{x} by the normal algebraic method.
(iv) Determine y and x by 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
d y
dt
+ x = 1
dx
dt
− y + 4e
t
= 0
given that at t = 0, x = 0 and y = 0.
Using the above procedure:
(i) L
d y
dt
+ L{x} = L{1}
(1)