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)