Differential Equation with a Discontinuous Forcing Function

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Consider the equation , where is a square-wave step function and is the oscillation of a spring-mass system in resonance with the square-wave forcing function. The graph of is drawn in purple and that of in blue. Using Laplace transforms, this solution is more compact than using a Fourier series expansion of the forcing function.


The first three terms of the Laplace transform of the homogeneous solution for are: . The Laplace transform of the forcing function is . The phase synchronization between input and output gives rise to resonance.


Contributed by: Stephen Wilkerson (March 2011)
(United States Military Academy West Point, Department of Mathematics)
Open content licensed under CC BY-NC-SA



This example comes from [1], Section 5.6, Differential Equations with Discontinuous Forcing Functions.


[1] J. R. Brannan and W. E. Boyce, Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications, New York: John Wiley and Sons, 2010.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.