
What is meant by Laplace Transform?  
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Definition of Laplace Transform and give the basic Laplace Transforms and its application.  
MathematicsLaplaceTransform  
 
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 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 PierreSimon Laplace, who used a similar transform(now called ztransform) in his work on probability theory. Basic Laplace Transforms: [$$:] \begin{align} & f(t) & L[f(t)]\\ 1.\; & 1 & \frac{1}{s} \\ 2.\; & e^{at} & \frac{1}{sa} \\ 3.\; & t^n, n=1,2,3... & \frac{n!}{s^{n+1}} \\ 4. \; & t^p, p>1 & \frac{\Gamma{(p+1)}}{s^{p+1}} \\ 5.\; & \sqrt{t} & \frac{\sqrt{\pi}}{2s^{\frac{3}{2}}} \\ 6.\; & t^{n1/2} , n=1,2,3... & \frac{1.2.3..(2n1)\sqrt{\pi}}{2^ns^{n+1/2}} \\ 7. \; & sin(at) & \frac{a}{s^2+a^2} \\ 8. \; & cos(at) & \frac{s}{s^2+a^2} \\ \end{align} [/:$$]  
 
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