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.