Unit – 4

Laplace transformation

4.1 Laplace transformation and its use in solving differential equations

PERIODIC FUNCTIONS:

Let f (t) be a periodic function with Period T, then

Example1: Find the Laplace transform of the waveform

Solution: 

Example2: Find the Laplace transform of the function (Half wave rectifier)

Solution:

Example3: Find the Laplace transform of the periodic function (saw tooth wave)

Solution:

4.2 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:

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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:

4.3 Integral equation

Example1: Evaluate 

Solution:   

Example2: Evaluate 

Solution:  

Example3: Evaluate 

Solution:  

Text Book:

1) Calculus: Gorakh Prasad

2) Advance Engineering Mathematics – E. Kreyszig, John Wiley & Sons Inc.