UNIT–5
Inverse Laplace Transform
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.
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-
We can solve the linear differential equations with constant coefficients without finding general solution and arbitrary constant, by using Laplace transform.
Example: Solve the following initial value problem by using convolution-
y’’ + y = sin 3t
y(0) = 0, y’(0) = 0
Sol.
y’’ + y = sin 3t
Taking Laplace transform of both sides, we have-
On putting the given values, we get-
On taking the inverse transformation, we get-
Example: Solve the following initial value problem by using Laplace transform-
Sol.
We can write this equation as-
y’’+2y’+2y = 5 sin t
Taking the Laplace transformation of both sides, we have-
Putting the given values, we get-
Resolving into partial fractions-
On taking inverse transform, we get-
