Module – 5

Laplace transformation

5.1 Laplace transformation and its use in getting solution to differential equations

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:

5.2 Convolution

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:

5.3 Integral Equations

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.