Inverse Laplace transform IMP Ques and Solutions

here you will get all imp Questions And thier Solution Also ....

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called the z-transform) in his work on probability theory.[2] The current widespread use of the transform (mainly in engineering) came about during and soon after World War II [3] although it had been used in the 19th century by Abel, Lerch, Heaviside, and Bromwich.

The early history of methods having some similarity to Laplace transform is as follows. From 1744, Leonhard Euler investigated integrals of the form

{displaystyle z=int X(x)e^{ax},dxquad {text{ and }}quad z=int X(x)x^{A},dx} z=int X(x)e^{ax},dxquad {text{ and }}quad z=int X(x)x^{A},dx
as solutions of differential equations but did not pursue the matter very far.[4]

Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

{displaystyle int X(x)e^{-ax}a^{x},dx,} int X(x)e^{-ax}a^{x},dx,
which some modern historians have interpreted within modern Laplace transform theory.[5][