UNIT 2

Laplace Transform

Q1) Find the Laplace transform of (1 + cos 2t)

A1)

So that-

Q2) Find the Laplace transform of

A2) Here-

So that-

As we know that-  

So that-

Hence-

Q3) Find the Laplace transform of the following function-

A3) The given function f(t) can be written as-

So that, by definition,

Q4) Find the Laplace transform of the following function (Half-wave rectifier)-

A4) By the definition-

f(t) is a periodic function and

So that-

As we know that-

Q5) Find the Laplace transform of the periodic function-

A5) By the definition-

Q6) Find the Laplace transform of .

A6) Here-

Now-

Q7) Find the Laplace transform of the following function by using unit step function-

A7)

Since

Q8) Find the Laplace transform of

Where-

A8) Here we are given—

As

Q9) Evaluate-

1.

A9)1. As we know that- 

So that-

2. As we know that-

Q10) Use Laplace transform method to solve the following equation-

A10) 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-

Q11) Find the solution of the initial value problem by using Laplace transform-

A11) Here we have-

Taking Laplace transform, we get-

Putting the given values, we get-

On inversion, we get-

4

Now-