Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

6: The Laplace Transform

( \newcommand{\kernel}{\mathrm{null}\,}\)

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.

  • 6.1: 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.
  • 6.2: 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.
  • 6.3: Convolution
    The Laplace transformation of a product is not the product of the transforms. Instead, we introduce the convolution of two functions of t to generate another function of t.
  • 6.4: Dirac Delta and Impulse Response
    Often in applications we study a physical system by putting in a short pulse and then seeing what the system does. The resulting behavior is often called impulse response.
  • 6.5: Solving PDEs with the Laplace Transform
    The Laplace transform comes from the same family of transforms as does the Fourier series, to solve partial differential equations (PDEs). It is therefore not surprising that we can also solve PDEs with the Laplace transform.
  • 6.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.


This page titled 6: The Laplace Transform is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jiří Lebl via source content that was edited to the style and standards of the LibreTexts platform.

  • Was this article helpful?

Support Center

How can we help?