Module-1
Laplace Transform
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.
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-
Properties and theorems of LT
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, by definition,
Existence theorem-
The Laplace transform of f(t) exists for s>a if –
- F(t) is continuous
is finite.
The above conditions are not necessary but sufficient.
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-
Laplace transform of the function 
multiplied by t
If 
, then-
Proof: 
Differentiate w.r.t. x, we get-
Similarly-
And
Example-4: Find the Laplace transform of 
.
Sol. Here-
Now-
We can solve the initial value problems directly without first finding a general solution by using the Laplace transform method for solving initial value problems.
Example: solve the initial value problem-
Sol.
Take the Laplace transform of the given differential equation, we get-
Using the initial conditions, we get-
On taking Laplace inverse of Y(s), we get-
Example: using Laplace transform find the solution of the following initial value problem-
y’’ + 25y = 10 cos 5t, y(0) = 2, y’(0) = 0
Sol.
Here we have-
y’’ + 25y = 10 cos 5t, y(0) = 2, y’(0) = 0
On taking Laplace transform of the given differential equation and using initial conditions, we get-
Example: using Laplace transform find the solution of 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-
Example: Solve the following initial boundary value problem-
Sol.
Taking Laplace to transform with respect to ‘t’ and using-
We get-
Or
It is a linear ordinary differential equation and its solution is given by-
When x = 0, 
Hence C = 
So that-
Now taking Laplace inverse of 
, we get-
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 a unit step function and find it's 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 
Dirac-delta function-
The impulse function is also known as the Dirac-delta 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-
1. 
Sol.1. As we know that- 
So that-
2. As we know that-
Example-2: 
Sol. 
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-
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
We can find the inverse Laplace transform by using the 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 the Laplace transform of both sides of the given differential equation.
2. Transpose the terms with a 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-
The solution of simultaneous differential equations by using Laplace transform-
Example: 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.
Reference Books
1. B.S. Grewal: Higher Engineering Mathematics; Khanna Publishers, New Delhi.
2. B.V. Ramana: Higher Engineering Mathematics; Tata McGraw- Hill Publishing Company Limited, New Delhi.
3. Peter V.O’ Neil. Advanced Engineering Mathematics, Thomas ( Cengage) Learning.
4. Kenneth H. Rosem: Discrete Mathematics its Application, with Combinatorics and Graph Theory; Tata McGraw- Hill Publishing Company Limited, New Delhi
5. K.D. Joshi: Foundation of Discrete Mathematics; New Age International (P) Limited,
Publisher, New Delhi.
