Table of contents
1.
Introduction
2.
Inverse Laplace Transform
2.1.
Transformation Table
2.2.
Properties
3.
Problems
4.
FAQs
5.
Key Takeaways
Last Updated: Mar 27, 2024

Inverse Laplace Transform

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Introduction

The Laplace Transform and Inverse Laplace Transform are powerful tools for solving non-homogeneous linear differential equations. In this article, we will learn what inverse Laplace transform does. We will also discuss the transformation formulas, some of their properties, and some questions to help you solve questions on inverse Laplace transform.

Inverse Laplace Transform

In Laplace Transform, we transformed a given derivative function with variable ‘t’ into a complex function with variable ‘s.’ Now, in Inverse Laplace Transform, we transform the complex function with variable ‘s’ back into a function with variable ‘t.’

The above statement can be expressed mathematically as

L{f(t)} → F(s) Laplace Transform

L-1{F(s)} → f(t) Inverse Laplace Transform

Now, you would be wondering that if we are taking the Laplace transform of a function and then we again take the inverse Laplace transform of the resultant function, then what is the point of transforming it in the first place? You will realize it when you are stuck in solving very tedious differential equations. The Laplace transform converts a somewhat differential calculation into a simple algebraic calculation. And in the end, we extract the equation in terms of its original variable using inverse Laplace transform.

We need to be well-versed with the Laplace transform to perform the inverse Laplace transform. We make use of the Laplace transform table only to get the inverse. We check that the given F(s) is of which form in the right-hand side of the Laplace transform table, and the result will be the left-hand side of the identity with which it matches. If we remember the Laplace transform table, then we will also be able to perform the inverse Laplace transform.

Transformation Table

Source

In the above table, the Inverse Laplace Transform is given on the left side, and Laplace Transform is shown on the right side. This table contains the most commonly used transformations but there are many other transformations as well.

Properties

Some essential properties which come in handy while solving problems on Inverse Laplace Transform are given below.

  • Additive Property
  • Linearity Property

    Where ‘a’ is some constant.
  • First Shift Theorem

    Where ‘a’ is some constant.
  • Second Shift Theorem

Problems

We will now solve some problems to help you better understand how to solve the inverse Laplace transform.

  1. Find the inverse Laplace Transform of ( 8 - 3s + s2)/s3
    L-1{F(s)} = L-1{ ( 8 - 3s + s2)/s3 }
    L-1{F(s)} = 8 L-1{ 1 / s3} - 3 L-1{ 1/s2 } + L-1{ 1/s }
    L-1{F(s)} = 8( t3-1 / 2! ) - 3( t2-1 / 1! ) + 1 [Using property 1 and 3]
    L-1{F(s)} = 4t2 - 3t + 1
  2. Find the Inverse Laplace Transform of [5 / (s-2)] - [4s / (s2 + 9)]
    L-1{F(s)} = 5 L-1{1 / (s-2)} + 4 L-1{s / (s2 + 9)}
    L-1{F(s)} = 5 e2t L-1{ 1/s } + 4 L-1{s / (s2 + 32)} [Using First Shift Theorem]
    L-1{F(s)} = 5 e2t + 4 Cos(3t) [Using property number 1 and 8]
  3. Find the Inverse Laplace Transform of (s-5)/(s2 + s - 6)
    L-1{F(s)} = L-1{(s-5)/(s2 + s - 6)}
    L-1{F(s)} = L-1{(s-5)/(s+3)(s-2)}
    Now, using partial fraction we can write the above equation as
    L-1{F(s)} = L-1{ [(8 / 5) / (s+3)] - [(3 / 5) / (s-2)] }
    L-1{F(s)} = (8 / 5)L-1{1 / (s+3)}  - (3 / 5)L-1{1 / (s-2)} [Using property 2]
    L-1{F(s)} = (8 / 5)e-3t - (3 / 5)e2t

FAQs

  1. What is inverse Laplace transform?
    It transforms a complex function with variable ‘s’ into a function with variable ‘t.’
  2. What is the linearity property?
    Linearity property states that inverse of a.F(s) is the same as ‘a’ times the inverse of F(s)
  3. What is the additive property of Inverse Laplace Transform?
    It states that the inverse of {F(s) + G(s)} will be the inverse of F(s) added to the inverse of G(s).
  4. What is the first shift theorem in Inverse Laplace Transform?
    According to the first shift theorem, the inverse of F(s-a), where a is some constant, is eat times the inverse of F(s).
  5. What is the second shift theorem?
    The Second shift theorem states that if there is a term of esT in the numerator of complex function, we find the inverse of the remaining function and then replace ‘t’ with ‘t-T’ in the resulting equation.

Key Takeaways

In this article, we have extensively discussed the Inverse Laplace Transform topic. We hope that this blog has helped you enhance your engineering mathematics knowledge. If you would like to learn more, check out our other articles.

  1. Laplace Transform
  2. Laplace transform of a Unit Step function
  3. Laplace Transform Properties
  4. Arithmetic Progression
  5. Mean, Variance, and Standard Deviation
  6.  Initial Value Theorem

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