Tutorial 7: Laplace Transform Worksheet
Solving an ODE using the Laplace Transform
1. Show that L(cos 2t) =
s
.
s2 +4
2. Use partial fractions to write
(s2
1 s
1 s
s
=
−
.
2
2
+ 1)(s + 4)
3 s + 1 3 s2 + 4
3. Solve x00 + x = cos(2t) with x(0) = 0 and x0 (0) = 1.
The Heaviside function
(
0, t < 0,
The Heaviside function, u(t), is defined as u(t) =
1, t ≥ 0.
1. Sketch u(t). Sketch f (t) = u(t − π) sin t. Sketch g(t) = u(t − 1) − u(t − 5).
1
2. Show that L(u(t − a)) =
e−as
s .
Note 1 This can be generalized: L(f (t − a)u(t − a)) = e−as F (s).
Note 2 Note that L(1 − cos t) =
1
s(s2 +1)
(check this later).
3. Solve x00 + x = f (t), x(0) = 0, x0 (0) = 0, where f (t) = 1 for 1 ≤ t < 5 and f (t) = 0 otherwise.
You’ll need Note 1 and Note 2.
Transfer functions give an algebraic dependence of the output based on the input
Consider Lx = f (t) with L a constant coefficient differential operator, with all initial conditions
0. Taking the Laplace Transform gives A(s)X(s) = F (s), so that X(s) = H(s)F (s) for any input
f (t). This suggests that x(t) can be found by multiplying F (s) by H(s) in the frequency-domain
and subsequently taking the inverse Laplace Transform.
1. Find the transfer function for the ODE x00 + ω02 x = f (t), assuming all initial conditions are 0.
2. Suppose f (t) = 1. Use the transfer function from above to find x(t).
2