8: The Laplace Transform
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.
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- 8.1: 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.
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- 8.2: 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.
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- 8.E: The Laplace Transform (Exercises)
- These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.
Contributors and Attributions
- Jiří Lebl ( Oklahoma State University ).These pages were supported by NSF grants DMS-0900885 and DMS-1362337.