Unit-2
Laplace Transformation
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 which 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 transforms-
The Laplace transform of f(t) exists for s>a, if
1. f(t) is a continuous function.
2. 
is finite.
Important formulae-
1. 
2. 
3. 
4. 
5. 
6. 
7. 
Example-1: Find the Laplace transform of the following functions-
1. 
2. 
Sol. 1.
Here 
So that we can write it as-
Now-
2. Since 
Or 
Now-
Example-2: Find the Laplace transform of (1 + cos 2t)
Sol. 
So that-
Inverse Laplace transforms-
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.
Important formulae-
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
Conditions for existence-
The Laplace transform of f(t) i.e. 
exists for s > a, if-
1. f(t) is continuous
2. 
is finite
Note- the above conditions are sufficient not necessary.
1. Linearity property-
Let a and b be any two constants and 
, 
any two functions of t, then-
Proof:
Hence proved.
2. First shifting property (Theorem)- If 
Proof: By definition-
Let (s – a) = r
Hence proved.
We can find the following results with the help of the above theorem-
1. 
2. 
4. 
5. 
6. 
7. 
Here s>a in each case.
Example-1: Find the Laplace transform of t sin at.
Sol. Here-
Example-2: Find the Laplace transform of 
Sol. Here-
So that-
As we know that- 
So that-
Hence-
Example-3: Find the Laplace transform of the following function-
Sol. The given function f(t) can be written as-
So that, by definition,
Change of scale property-
Example: Find-
Given that,
Sol.
By using change of scale property-
Unit step function
The unit step function u(t – a) is defined as-
Laplace transform of unit functions-
Example-1: Express the function given below in terms of unit step function and find its Laplace transform as well-
Sol. Here we are given-
So that-
Example-2: Find the Laplace transform of the following function by using unit step function-
Sol. 
Since 
Second shifting theorem-
If 
Then,
Proof: 
Example-3: Find the Laplace transform of 
Where-
Sol. Here we are given—
As
Unit impulse function
Impulse- When a large force acts for a short time, then the product of the force and the time is called impulse.
The unit impulse function is the limiting function.
= 0, otherwise
The unit impulse function can be defined as-
And
Laplace transform of unit impulse function-
We know that-
Mean value theorem- 
As 
, then we get- 
When 
then we have 
Example-1: Evaluate-
Sol.1. As we know that- 
So that-
2. As we know that-
Example-2: 
Sol. 
Laplace transform of periodic functions-
If f(t) is a periodic function with period T, which means f(t + T) = f(t), then-
Proof: we have-
Put t = u + T in the second integral, t = u + 2T in third integral and so on…
Then-
Laplace transform of the derivative of f(t)-
Here 
Proof: by the definition of Laplace transform-
On integrating by parts, we get-
Since 
Then-
So that-
Laplace transform of integral of f(t) -
Proof: Suppose
We know that-
So that-
Putting the values of 
and 
, we get-
Multiplication by 
-
If 
Then
Proof:
We have
Where the double integral is taken over the infinite region in the first quadrant lying between the lines x = 0 and x = t.
If we change the order of integration, the above integrals changes to-
Hence proved.
Example: Apply convolution theorem to evaluate-
Sol.
Since
By convolution theorem, we get-
Because- f(u) = cos au, g(t-u) = cos b(t – u)
Inverse Laplace transforms-
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.
Important formulae-
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
Example: Find the inverse Laplace transform of the following functions-
1. 
2. 
Sol.
1.
2.
Example: Find the inverse Laplace transform of-
Sol.
Multiplication by ‘s’ -
Example: Find the inverse Laplace transform of-
Sol.
Division by s-
Example: Find the inverse Laplace transform of-
Sol.
Inverse Laplace transform of derivative-
Example: Find 
Sol.
Inverse Laplace transform by using partial fraction and convolution theorem
We can find the inverse Laplace transform by using partial fractions method described below-
Example: Find the Laplace inverse of-
Sol.
We will convert the function into partial fractions-
Example: Find the inverse transform of-
Sol.
First we will convert it into partial fractions-
Inverse Laplace transform by convolution theorem-
According to the convolution theorem-
Example: Find
Sol.
Therefore by the convolution theorem-
Step by step procedure to solve a linear differential equation by using Laplace transform-
1. Take Laplace transform of both sides of the given differential equation.
2. Transpose the terms with negative sign to the right.
3. Divide by the coefficient of 
, getting 
as a known function of s.
4. Resolve the function of s into partial fractions and take the inverse transform of both sides.
We will get y as a function of t. Which is the required solution.
Example-1: 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-
Example-2:Use Laplace transform method to solve the following equation-
Sol.
Here, taking the Laplace transform of both sides, we get
It becomes-
On inversion, we get-
Example-3:Use Laplace transform method to solve the following equation-
Sol. Here we have-
Taking Laplace transform of both sides, we get-
We get on putting given values-
On inversion, we get-
Example-4: Find the solution of the initial value problem by using Laplace transform-
Sol. Here we have-
Taking Laplace transform, we get-
Putting the given values, we get-
On inversion, we get-
4
Now-
