Explanation of Laplace transform

Question

In my Differential equations course I have topic on Laplace transform, I didn't understand the concept of Laplace transform. I searched it but I don't understand. I referred to one video https://youtu.be/6MXMDrs6ZmA Can someone please help me with this?

Answer 1

Advanced Differential Equation By MD Rai singhania

Refer this book for Laplace Transformation . If you are comfortable in HINDI do watch enter link description here

Answer 2

There is nothing mysterious in the "concept" of Laplace transform.

The Laplace transform is a particular example of integral transform. They are many other integral transforms, for example Fourier transform, Mellin transform, Hankel transform, Hilbert transform, etc. Many Great Mathematicians invented their own Integral transform.

The idea behind is to apply an operator, say an integral, to a function in order to transform it into another function. If you do this for all functions in an equation, the equation is transformed into another equation. Sometimes, but not always, the transformed equation is much easier to solve that the original equation. So, the transformed equation is solved and the solution(s) obtained have to be treated with the inverse operator to recover the solution(s) of the original equation.

For example, you transform the original equation thanks to Laplace transform. You solve the new equation. You transform the solution(s) thanks to the Inverse Laplace transform. This gives the solution(s) of the original equation.

Of course, to carry out those transformations, there is no need for heavy calculus to compute the integrals. One simply find the transformed functions in available Tables of Integral transforms. In fact, when proceeding with standard integral transform, one avoid the difficult part of the job which was done a long time ago by the mathematicians who computed the tables of integral transforms and inverse integral transforms.

Sorry, the above talk is nothing more than popularization. I beg indulgence from the purists.