5: The Laplace Transform
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The Laplace transform can also be used to solve differential equations and reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.
- 5.1: The Laplace Transform
- The Laplace transform turns out to be a very efficient method to solve certain ODE problems. In particular, the transform can take a differential equation and turn it into an algebraic equation. If the algebraic equation can be solved, applying the inverse transform gives us our desired solution.
- 5.2: Transforms of Derivatives and ODEs
- The procedure for linear constant coefficient equations is as follows. We take an ordinary differential equation in the time variable t . We apply the Laplace transform to transform the equation into an algebraic (non differential) equation in the frequency domain. We solve the equation for X(s) . Then taking the inverse transform, if possible, we find x(t). Unfortunately, not every function has a Laplace transform, not every equation can be solved in this manner.
- 5.3: Convolution
- This page discusses the use of inverse Laplace transforms and convolution in solving ordinary differential and Volterra integral equations. It highlights the simplification of computations through these methods. An example is provided where a differential equation involving an integral is transformed into the frequency domain, resulting in the expression X(s)=s−1s2−2. The final solution is obtained as x(t)=cosh(√2t)−1√2sinh(√2t).
Contributors and Attributions
- Jiří Lebl (Oklahoma State University).These pages were supported by NSF grants DMS-0900885 and DMS-1362337.