Laplace-transform - Notes on Laplace transform In a rst...

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Notes on Laplace transform In a first approximation, Laplace transform is quite analogous to logarithms. The main property of logarithms is that ”taking logarithms” we transform products into sums and powers into products. log( a · b ) = log( a ) + log( b ) log( a b ) = b log( a ) This allows us to transform equations involving powers into equations involving prod- ucts, which may be more approachable. For example, if given the equation 7 x = 5 we realize that this is equivalent to log(7 x ) = log(5), which using the properties of loga- rithms is equivalent to x log(7) = log(5); x = log(5) / log(7) . 8270874753 Note that in the last step, we had to resort to a calculator that gives us the logarithms. (In older days, you had to resort to looking them up in a table.) Note that an important properties of logarithms is that knowing the logarithm of a number identifies the number. Laplace transform plays a similar role in differential equations. Taking Laplace trans- form of a an initial value problem for a linear diffferential equation with constant coefficients reduces the problem to an algebraic problem. Remember that a differential equation is an equation whose unknown is a function. Laplace transform will be a gadget that given a function produces another function.
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