ENGINEERING MATHEMATICS II
[FEM 1023]
CHAPTER 3
LAPLACE TRANSFORM FOR
ORDINARY DIFFERENTIAL
EQUATIONS
Overview
3.1. Laplace Transform
3.2. Inverse Laplace Transform
3.3. Translation Theorem
3.4. Method of Solution
3.1. Laplace Transform
Learning Outcomes
At the end of this section you should be able to:
1) Define Laplace Transform
2) Find the Laplace Transform of different type of functions
3.1. Laplace Transform
Pierre-Simon Laplace
(1749–1827)
3.1. Laplace Transform
Why?
There are Differential Equations that will be difficult
for us to solve just knowing whatever methods we
have learnt so far.
For example:
Mass/spring system
Series electrical circuit
3.1. Laplace Transform
The Laplace transform is used to produce an easily
solvable algebraic equation from an ordinary
differential equation.
Laplace Transform is a special type of integral
transform, it has many interesting properties that
make it very useful in solving linear IVPs.
It has important applications in Mathematics, Physics
and Engineering.
3.1. Laplace Transform
Definition
Integral Transform
Then, the improper integral
is an integral transform.
3.1. Laplace Transform
Definition
Laplace Transform
Then the integral,
3.1. Laplace Transform
Notations
Lowercase letter denote the function being
transformed.
Capital letter denote its Laplace transform.
3.1. Laplace Transform
Example 1
Solution
3.1. Laplace Transform
3.1. Laplace Transform
Example 2
Solution
3.1. Laplace Transform
3.1. Laplace Transform
Example 3
Solution
3.1. Laplace Transform
3.1. Laplace Transform
Transform of Some Basic Functions
3.1. Laplace Transform
For a linear combination of functions
Example
3.1. Laplace Transform
Example 4
Evaluate
Example 5
Evaluate
Example 6
− 1 0  t  1
f (t ) = 
t 1
1