Continue changing w
to get even closer to k. What happens to the amplitude of the solution?
What happens to the frequency of the beats? What sort of solution function
do you seem to be approaching? You are exploring the onset of resonance,
which means driving the system at its natural frequency.
Redefine the initial value
problem so that w = k, that is, so the system is being driven at
its natural frequency:
y'' + k2 y = F0 cos kt, y(0) = 0, y'(0)
= 0.
Find the symbolic solution of this initial value problem. You may use your
helper application.
Plot the solution and its
trajectory in the phase plane for k = 5 and F0 = 1.
What feature of the symbolic solution explains
the growing amplitudes? Will the growth in amplitude ever stop (as it did
in the beats setting)? Explain.