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.