Module-1

Laplace Transform

Question-1: Define Laplace transform and its conditions for existence.

Sol.

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.

Conditions for the existence of Laplace transform-

The Laplace transform of f(t) exists for s>a, if

1. f(t) is a continuous function.

2. is finite

Question-2: Find the Laplace transform of the following function-

Sol.

Here

So that we can write it as-

Now-

Question-3: Find the Laplace transform of (1 + cos 2t)

Sol.

So that-

Question-4: define the first shifting property of Laplace transform

Sol.

(Theorem)- If

Proof: By definition-

Let (s – a) = r

Hence proved.

Question-5: Find the Laplace transform of

Sol. Here-

So that-

As we know that-  

So that-

Hence-

Question-6: Find the Laplace transform of the following function-

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

So, by definition,

Question-7: Find the Laplace transform of .

Sol. Here-

Now-

Question-8: using Laplace transform find the solution to the following initial value problem-

y’ + y =

Sol. On taking the Laplace transform and using the initial conditions, we get-

Thus

On breaking into partial fractions, we get-

Question-9: Express the function given below in terms of a unit step function and find it's Laplace transform as well-

Sol. Here we are given-

So that-

Question-10: Evaluate-

1.

Sol.1. As we know that- 

So that-

2. As we know that-

Question-11: Find the inverse Laplace transform of-

Sol.

Question-12: Find

Sol.

Question-13: Use Laplace transform method to solve the following equation-

Sol. 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-

Question-14: solve the following differential equation by using Laplace transform-

Here D = d/dt and

Sol.

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.

Module-1

Laplace Transform

Question-1: Define Laplace transform and its conditions for existence.

Sol.

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.

Conditions for the existence of Laplace transform-

The Laplace transform of f(t) exists for s>a, if

1. f(t) is a continuous function.

2. is finite

Question-2: Find the Laplace transform of the following function-

Sol.

Here

So that we can write it as-

Now-

Question-3: Find the Laplace transform of (1 + cos 2t)

Sol.

So that-

Question-4: define the first shifting property of Laplace transform

Sol.

(Theorem)- If

Proof: By definition-

Let (s – a) = r

Hence proved.

Question-5: Find the Laplace transform of

Sol. Here-

So that-

As we know that-  

So that-

Hence-

Question-6: Find the Laplace transform of the following function-

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

So, by definition,

Question-7: Find the Laplace transform of .

Sol. Here-

Now-

Question-8: using Laplace transform find the solution to the following initial value problem-

y’ + y =

Sol. On taking the Laplace transform and using the initial conditions, we get-

Thus

On breaking into partial fractions, we get-

Question-9: Express the function given below in terms of a unit step function and find it's Laplace transform as well-

Sol. Here we are given-

So that-

Question-10: Evaluate-

1.

Sol.1. As we know that- 

So that-

2. As we know that-

Question-11: Find the inverse Laplace transform of-

Sol.

Question-12: Find

Sol.

Question-13: Use Laplace transform method to solve the following equation-

Sol. 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-

Question-14: solve the following differential equation by using Laplace transform-

Here D = d/dt and

Sol.

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.

Module-1

Module-1

Laplace Transform

Question-1: Define Laplace transform and its conditions for existence.

Sol.

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.

Conditions for the existence of Laplace transform-

The Laplace transform of f(t) exists for s>a, if

1. f(t) is a continuous function.

2. is finite

Question-2: Find the Laplace transform of the following function-

Sol.

Here

So that we can write it as-

Now-

Question-3: Find the Laplace transform of (1 + cos 2t)

Sol.

So that-

Question-4: define the first shifting property of Laplace transform

Sol.

(Theorem)- If

Proof: By definition-

Let (s – a) = r

Hence proved.

Question-5: Find the Laplace transform of

Sol. Here-

So that-

As we know that-  

So that-

Hence-

Question-6: Find the Laplace transform of the following function-

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

So, by definition,

Question-7: Find the Laplace transform of .

Sol. Here-

Now-

Question-8: using Laplace transform find the solution to the following initial value problem-

y’ + y =

Sol. On taking the Laplace transform and using the initial conditions, we get-

Thus

On breaking into partial fractions, we get-

Question-9: Express the function given below in terms of a unit step function and find it's Laplace transform as well-

Sol. Here we are given-

So that-

Question-10: Evaluate-

1.

Sol.1. As we know that- 

So that-

2. As we know that-

Question-11: Find the inverse Laplace transform of-

Sol.

Question-12: Find

Sol.

Question-13: Use Laplace transform method to solve the following equation-

Sol. 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-

Question-14: solve the following differential equation by using Laplace transform-

Here D = d/dt and

Sol.

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.