Inverse Laplace Transform: Overview, Questions, Preparation

The Laplace Transform is named after the French mathematician L. S. P. Laplace (1749-1827). As with other transformations, the Laplace Transform transforms one signal to another depending on one or a few fixed laws. The simplest approach to explain an equation is to turn it into an algebraic equation. 

The Laplace Transform is the integral transform of the specified derivative function to turn into a complex function with real variables. For a certain t, let f(t) be given (to fulfil the conditions mentioned above later on).

The Laplace Transform of an equation is described by the equation ‘f(t)=f(s) whenever the integration converges to zero.’

Where a notation is simple, I would write our Laplace Transforms with capital letters, e.g. L(f; s) = F(s). 

The Laplace Transform we described corresponds to the Laplace Transform available on only one hand. There is a two-sided representation of the integral that moves from negative infinity to positive infinity. 

If the function y(a) is a continuous function, which is defined on [0, ∞) and also satisfies L[y(a)](b) = Y(b), then it is a Laplace Transform of Y. (b). 

You may use a piecewise continuous function over a given interval if the complementary function is continuous everywhere. 

L-1[Y(b)] (a) 

The integral determines the Laplace Transform where Y(b) is the Y(a) of the function specified on [o, ∞].

To do the inverse transformation, use the table shown below.

Inverse Laplace Transform in Class 12

This concept is taught under the chapter “Inverse Trigonometric Functions.” You will learn about the basics and formulas for inverse Laplace Transform. The weightage of this chapter is 8 marks.

Find the inverse transform of each of the following:

Y(b)= 6/2b -1/(b−4) – 4/(b−2)

Solution:

Step 1: The first term is a constant as we can see from the first term’s denominator.

Step 2: Before taking the inverse transform, let’s take the factor 6 out, so the correct

the numerator is 6.

Step 3: The second term has an exponential t = 4.

Step 4: The numerator is perfect.

Step 5: The third term is also an exponential, t= 2.

Step 6: Now, before taking the inverse transforms, we need to factor out 4 first.

Y(b) = 6.1/2b -1/(b−4) – 4.1/(b−2)

y(a) = 6(2) – e^4a +4^(e2t)

= 12 – e^4a +4e^2t

Q: What is s in Laplace transformation?

Q: What does the inverse Laplace Transform do?

Q: Is the Laplace Transform nonlinear?

Q: Can you multiply the Laplace Transform?

Q: Why are the Laplace Transforms important?