Module – 5
Laplace transformation
Definition:Let 
be function defined for all positive values of t, then 
Provided the integral exists, is called the Laplace Transform of 
. It is denoted as -
Important Formulae:
4. 
5. 
6. 
7. 
Example1. Find the Laplace transform of 
defined as
Solution:
Example2. From the first principle, find the Laplace transform of (1 + cos 2 t)
Solution: Laplace transform of (1 + cos 2 t)
Example3. Find the Laplace transform of
Solution:
Now we obtain f (t) when F (s) is given, then we say that inverse Laplace transform of F (s) is f (t).
Where,
is called the inverse Laplace transform operator.
From the application point of view, the inverse Laplace transform is very useful.
IMPORTANT FORMULAE:
4. 
5. 
6. 
7. 
8. 
9. 
10.
11.
12.
13.
14.
15.
16.
17.
MULTIPLICATION by s: 
DIVISION by s:
Example1: Find the inverse Laplace Transform of
Solution:
Example2: Find the inverse Laplace Transform of
Solution:
Example3: Find the inverse Laplace Transform of function by MULTIPLICATION by s
Solution:
Example4: Find the inverse Laplace Transform of function by Division by s
Solution:
Example1: Evaluate 
Solution: 
Example2: Evaluate 
Solution: 
Example3: Evaluate 
Solution: 
References
1. Ordinary and Partial Differential equations by J. Sihna Ray and S Padhy, Kalyani Publishers
2. Advance Engineering Mathematics by P.V.O’NEIL, CENGAGE
3. Ordinary Differential Equation by P C Biswal , PHI secondedition.
4. Engineering Mathematics by P. S. Das & C. Vijayakumari, Pearson. N.B:Thecourseisof3creditwith4contacthours.
