Differential Equation with a Discontinuous Forcing Function

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.

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

Reference

[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.