Laplace transform is yet another operational tool for solving constant coefficients linear differential equations. The process of solution consists of three main steps:

      - The given "hard" problem is transformed into a "simple" equation.

      - This simple equation is solved by purely algebraic manipulations.

      - The solution of the simple equation is transformed back to obtain the solution of the given problem.

In this way the Laplace transformation reduces the problem of solving a differential equation to an algebraic problem.

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform(now called z-transform) in his work on probability theory.

Basic Laplace Transforms: [$$:] \begin{align} & f(t) & L[f(t)]\\ 1.\; & 1 & \frac{1}{s} \\ 2.\; & e^{at} & \frac{1}{s-a} \\ 3.\; & t^n, n=1,2,3... & \frac{n!}{s^{n+1}} \\ 4. \; & t^p, p>-1 & \frac{\Gamma{(p+1)}}{s^{p+1}} \\ 5.\; & \sqrt{t} & \frac{\sqrt{\pi}}{2s^{\frac{3}{2}}} \\ 6.\; & t^{n-1/2} , n=1,2,3... & \frac{1.2.3..(2n-1)\sqrt{\pi}}{2^ns^{n+1/2}} \\ 7. \; & sin(at) & \frac{a}{s^2+a^2} \\ 8. \; & cos(at) & \frac{s}{s^2+a^2} \\ \end{align} [/:$$]