Unit – 1

Laplace Transformation

Q1) What is Laplace transform?

A1)

Let f(t) be any function of t defined for all positive values of t. Then the Laplace transform of the function f(t) is defined as-

Provided that the integral exists, here ‘s’ is the parameter that could be real or complex.

The inverse of the Laplace transform can be defined as below-

Here

f(t) is called the inverse Laplace transform of

L is called the Laplace transformation operator.

Q2) Find the Laplace transform of the functions  

A2)

Here

So that we can write it as-

Now-

Q3) Find the Laplace transform of (1 + cos 2t)

A3)

So that-

Q4) What is the linearity property of LT?

A4)

Let a and b be any two constants and , any two functions of t, then-

Proof:

Hence proved.

Q5) Find the Laplace transform of t sin at.

A5)

Here-

Q6) Find the Laplace transform of the following function-

A6)

The given function f(t) can be written as-

So, by definition,

Q7) What do you understand by periodic function?

A7)

Suppose the function f(t) be periodic with period , then-

Similarly-

Now-

Put t = z + in the second integral of the above equation and t = z + 2 in the third integral and so on.

We get-

F(t) is periodic with period we can write-

Here is a geometric progression with a common ratio , and we know that the sum of infinite terms in G.P. Is given by
 

Then-

Q8) Define unit step function.

A8)

Unit step function

The unit step function u(t – a) is defined as-

Laplace transform of unit functions-

Q9) Find the Laplace transform of the following function by using unit step function-

A9)

Since

Q10) Evaluate-

A10)

As we know that-

Q11) Find the inverse Laplace transform of-

A11)

Q12) Find

A12)

Q13) Find the Laplace inverse of-

A13)

We will convert the function into partial fractions-

Q14) Find

A14)

Therefore by the convolution theorem-

Q15) Use Laplace transform method to solve the following equation-

A15)

Here we have-

Take Laplace transform of both sides, we get-

It becomes-

(

So that-

Now breaking it into partial fractions-

We get the following results on inversion-

Q16) Use Laplace transform method to solve the following equation-

A16)

Here we have-

Taking Laplace transform of both sides, we get-

We get on putting given values-

On inversion, we get-

Q17) solve the following differential equation by using Laplace transform-

Here D = d/dt and

A17)

Here we have-

Now multiply (1) by D+1 and (2) by D – 1 we get-

Now subtract (4) from (3), we get-

By taking Laplace to transform we get-

Put the value of in (1) we get-

By taking Laplace to transform we get-

Which is the required answer.

Q18) Find the inverse transform of-

A18)

First, we will convert it into partial fractions-