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Table of contents
1.
Introduction
2.
Laplace Transform
3.
Linearity Property of Laplace Transform
3.1.
Proof
4.
Numerical Examples
5.
Frequently Asked Questions 
6.
Conclusion
Last Updated: Mar 27, 2024

Linearity Property of Laplace Transform

Author Hari Sapna Nair
2 upvotes
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23 Jul, 2024 @ 01:30 PM

Introduction

Did you know that the graphs of current and voltage across a resistor, mass-weight, etc., exhibit a linearity property? This makes us wonder, what is the linearity property in Laplace transforms? 

 

In this blog, we will learn about the linearity property in Laplace transforms in detail and some numerical examples. Before we jump in, learn about what Laplace transforms are?

Laplace Transform

The Laplace transform is a mathematical tool that converts the differential equation in the time domain into the algebraic equations in the frequency domain(s-domain).

 

Mathematically, if x(t) is a time-domain function, then its bilateral Laplace transform is defined as given below −

The unilateral Laplace transform is defined as given below −

To learn more about Laplace Transform, check out our blog on Laplace Transform.

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Linearity Property of Laplace Transform

The linearity property of Laplace transforms states that the Laplace transform of the weighted sum of two signals is equal to the weighted sum of the individual sum Laplace transforms.


Consider the functions x1(t) and  x2(t) whose Laplace transform exists.

 

If  a  and  b  are constants, then, according to the linearity property of Laplace transform,

Proof

Now let's learn how we can prove the linearity property of Laplace transform.
 

According to the definition of Laplace transform, 

This is the same as,

Numerical Examples

Let's discuss some problems based on the linearity property of Laplace transform.

 

1. Find the Laplace transform of f(t) = 10t−5.
 

Solution:

L[10t − 5] = 10L(t) − 5L(1) [Using linearity property of Laplace transform]
 

The Laplace transform of the functions are as follows:-

10L(t) = 10(1/s2)

5L(1) = 5(1/s)

 

∴ L[10t − 5] = 10(1/s2) − 5(1/s) 

L[10t − 5] = 10/s2 − 5/s

L[10t − 5] = (10 - 5s)/s2 
 

2. Find the Laplace transform of x(t) = 2e−5tu(t) − 15e4tu(−t).

 

Solution:

L[2e−5tu(t) − 15e4tu(−t)] = 2L[e−5tu(t)] − 15L[e4tu(−t)] [Using linearity property of Laplace transform]


The Laplace transform of the functions are as follows:-

2L[e−5tu(t)] = 2/(s + 5)

15L[e4tu(−t)] = 15/(s - 4)

 

∴ L[2e−5tu(t) − 15e4tu(−t)] = 2/(s + 5) − 15/(s - 4)

L[2e−5tu(t) − 15e4tu(−t)] = (17s - 83)/(s2+s−20)
 

3. By using the linearity property of Laplace transform, show that

L(cosh at) = s/(s2−a2)

 

Solution:

cosh at = (eat + e−at)/2

L(cosh at) = L(eat)/2 + L(e-at)/2 
 

The Laplace transform of the functions are as follows:-

L(eat) = 1/(s - a)

L(e-at) = 1/(s + a)
 

∴ L(cosh at) = L(eat)/2 + L(e-at)/2 

L(cosh at) = (1/(s - a))/2 + (1/(s + a))/2

L(cosh at) = (1/(s - a) + 1/(s + a))/2

L(cosh at) = s/(s2−a2)

 

Hence Proved.

Frequently Asked Questions 

  1. Is Laplace transform a linear operator?
    Yes, the Laplace transform is a linear operator with the linearity property.
     
  2. Can the linearity property of Laplace transform be extended? 
    Yes, the linearity property of Laplace transform can be extended to more than two functions.
     
  3. Name some properties of Laplace transform.
    Some properties of Laplace transform are frequency shifting property, multiplication by time, time scaling property, time reversal property, etc.
     
  4. How can we prove the linearity property of Laplace transform?
    The linearity property of Laplace transform can be proved by using the definition of Laplace transform. 

Conclusion

In this article, we have extensively discussed the linearity property of Laplace transform along with some numerical examples.


Check out this problem - Subarray Sum Divisible By K

We hope that this blog has helped you enhance your knowledge regarding the linearity property of Laplace transform and if you would like to learn more, check out our article Inverse Laplace Transform and, Initial Value Theorem Do upvote our blog to help other ninjas grow. Happy Coding!

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