# Differential Equation with a Discontinuous Forcing Function

Differential Equation with a Discontinuous Forcing Function

Consider the equation u(t)+u(t)=g(t), 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.

2

d

d

2

t

2

π

g(t)

u(t)

g(t)

u(t)

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.

u(t)

-2s

s(+)

2

s

2

π

-s

s(+)

2

s

2

π

1

s(+)

2

s

2

π

-2s

s

-s

s

1

s

g(t)

u(t)