ENGI 3424
3 – Laplace Transforms
Page 3.01
3.
Laplace Transforms
In some situations, a difficult problem can be transformed into an easier problem, whose
solution can be transformed back into the solution of the original problem.
For example,
an integrating factor can sometimes be found to transform a non-exact first order first
degree ordinary differential equation into an exact ODE [Section 1.7]:
One type of first order ODE [not considered in this course] is the Bernoulli ODE,
y'
+
P y
=
R y
u
,
which, upon a transformation of the dependent variable, becomes a
linear ODE:
An
initial value problem
is an ordinary differential equation together with sufficient
initial conditions to determine all of the arbitrary constants of integration.
A Laplace
transform will convert some initial value problems into much easier algebra problems.
The solution of the original problem is then the inverse Laplace transform of the solution
to the algebra problem.
Uses of Laplace transforms include:
1)
the solution of some ordinary differential equations
2)
the solution of some integro-differential equations, such as
( )
( )
(
)
(
)
0
t
y t
g t
h t
x y x d
=
+
−
∫
x
.

ENGI 3424
3 – Laplace Transforms
Page 3.02
Contents
:
3.01
Definition, Linearity, Laplace Transforms of Polynomial Functions
3.02
Laplace Transforms of Derivatives
3.03
Review of Complex Numbers
3.04
First Shift Theorem, Transform of Exponential, Cosine and Sine Functions
3.05
Applications to Initial Value Problems
3.06
Laplace Transform of an Integral
3.07
Heaviside Function, Second Shift Theorem; Example for RC Circuit
3.08
Dirac Delta Function, Example for Mass-Spring System
3.09
Laplace Transform of Periodic Functions; Square and Sawtooth Waves
3.10
Derivative of a Laplace Transform
3.11
Convolution; Integro-Differential Equations; Circuit Example
Note
:
Sections 3.09, 3.10 and the latter part of 3.11 will be parts of this course only if
time permits.

ENGI 3424
3.01 – Definition; Polynomial Functions
Page 3.03
3.01
Definition, Linearity, Laplace Transforms of Polynomial Functions
If
f
(
t
)
is
defined on
t
> 0,
piece-wise continuous on
t
> 0
(that is, only a finite number of finite discontinuities) and
of exponential order
[ |
f
(
t
) |
<
k e
α
t
for all
t
> 0 and for some positive constants
k
and
α
],
then
exists and
( )
( )
0
lim
m
m
st
F s
e
f
t dt
→ ∞
−
=
∫
the Laplace transform of
f
(
t
)
is
( )
( )
( )
{
}
0
st
F s
e
f
t dt
f
t
∞
−
=
=
∫
L
Example 3.01.1
Find
L
{ 1 } .

ENGI 3424
3.01 – Definition; Polynomial Functions
Page 3.04
Example 3.01.2
Find
L
{
t
} .
Linearity Property of Laplace Transforms
L
{
a f
(
t
)
+
b g
(
t
) }
(where
a
,
b
are constants):

ENGI 3424
3.01 – Definition; Polynomial Functions
Page 3.05
Example 3.01.3
Find
L
{ 2
t
−
3 } .
Example 3.01.4
Find
L
{
t
n
} .