5Transform theory
IB Complex Methods
5.5 Solution of differential equations using the Laplace
transform
The Laplace transform converts ODEs to algebraic equations, and PDEs to
ODEs. We will illustrate this by an example.
Example. Consider the differential equation
t¨y − t ˙y + y = 0,
with y(0) = 2 and ˙y(0) = −1. Note that
L(t ˙y) = −
d
dp
L( ˙y) −
d
dp
(pˆy − y(0)) = −pˆy
0
− ˆy.
Similarly, we find
L(t¨y) = −p
2
ˆy
0
− 2pˆy + y(0).
Substituting and rearranging, we obtain
pˆy
0
+ 2ˆy =
2
p
,
which is a simpler differential equation. We can solve this using an integrating
factor to obtain
ˆy =
2
p
+
A
p
2
,
where A is an arbitrary constant. Hence we have
y = 2 + At,
and A = −1 from the initial conditions.
Example. A system of ODEs can always be written in the form
˙
x = M x, x(0) = x
0
,
where
x ∈ R
n
and
M
is an
n × n
matrix. Taking the Laplace transform, we
obtain
p
ˆ
x − x
0
= M
ˆ
x.
So we get
ˆ
x = (pI −M)
−1
x
0
.
This has singularities when p is equal to an eigenvalue of M.