5: The Laplace Transform

Introductory Differential Equations · LibreTexts

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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 \( X(s) = \dfrac{s-1}{s^2-2} \). The final solution is obtained as \( x(t) = \cosh(\sqrt{2}\, t) - \dfrac{1}{\sqrt{2}} \sinh(\sqrt{2}\,t) \).

    Contributors and Attributions

    This page titled 5: The Laplace Transform was last modified on Sun, 08 Dec 2024 18:06:58 GMT and is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Fatemeh Yarahmadi.