# Does ODE initial value problem produce beat or resonance phenomenon?

$$x''+9x=\sin(3t),$$

$$x(0)=x'(0)=0.$$

This question was asked on a test. We are allowed to solve differential equations with TI-89.

My steps:

1. Solve with TI-89, solution $$x(t) = \frac{1}{18} (\sin(3 t)-3 t \cos(3 t)) .$$
2. Plot the solution, and then look at the graph, and decide whether it's a beat or resonance.

Apparently, we are not allowed to solve the equation. How can i decide whether this IVP produce beat or resonance?

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You are forcing the system at the natural frequency of the system ($\omega_0 = \pm 3$), so it is unlikely to get a beat (amplitude modulation when the excitation frequency differs from the system modes).
Since there is a $t$ term in the response, the amplitude is unbounded. This is presumably what you call resonance.
The unforced system is $x'' = -3^2 x$. This has solutions $t \mapsto e^{\pm 3 i t}$. –  copper.hat Jul 9 '12 at 3:39