Understanding the Inverse Laplace Transform with Practical Examples

The Inverse Laplace Transform is an important mathematical tool used in engineering and physics, particularly in solving differential equations and analyzing linear time-invariant systems. In this article, we will explore how to apply the Inverse Laplace Transform to find the original function from its transformed form. We will specifically focus on the following example: finding the Inverse Laplace Transform of the function 1/s^2.

Problem Defined

Given the function
F(s) 1 / (s^2 25s)
our goal is to find its Inverse Laplace Transform.

Step-by-Step Solution

The first step in solving this problem involves simplifying the given function to a form that matches a known transform pair. We can achieve this through the partial fraction decomposition technique.

Let's rewrite the function as follows:

F(s) 1 / 25s 1/5 / (s 5)

In this step, we need to identify the constants, which are the roots of the denominator. To find these constants, we can set the denominator to zero:

s 5 0 → s -5

So, the function can be rewritten as:

F(s) 1/5 / (s 5)

Partial Fraction Decomposition

To decompose the fraction, we rewrite it using the constants we found:

F(s) 1/25 / (s 5)

Now, let's multiply both sides by 5s:

1 5 / (s 5)

To find the constants, we can substitute specific values for s. For instance:

If s 0, then
1 5 / 5 1

Similarly, if s 5, then
1 5 / 10 1/2

So, the constants are indeed 1/5 and 1/10, respectively. Therefore, the function can be written as:

F(s) 1/5 * 1 / (s 5)

Applying the Inverse Laplace Transform

Now that we have the simplified form of the function, we can apply the Inverse Laplace Transform to each term separately.

The Inverse Laplace Transform of 1 / s is 1.

The Inverse Laplace Transform of 1 / (s 5) is e^(-5t).

Thus, the Inverse Laplace Transform of F(s) is:

L^-1(1 / s^2 25s) 1/5 * 1 - 1/5 * e^(-5t)

Which simplifies to:

F(t) 1/5 - 5e^(-5t)

Conclusion

By following the steps outlined in this article, we have found the Inverse Laplace Transform of the given function. Understanding and applying Inverse Laplace Transforms is crucial in solving various real-world problems in engineering and physics. For further insights, join our Engineering Mathematics section.

References:

Partial fraction decomposition Inverse Laplace Transform formulas

Keywords: Inverse Laplace Transform, Partial Fraction Decomposition, Electrical Engineering