[Math] Show that there are two values of $\omega$ at which resonance occurs

Question

Consider a mass-spring system with $m = 1$, no damping, and $k = 4$. Find the general form of the mass’ motion if there is no forcing. Now suppose that we apply an external force $F(t) = 3\cos^3(\omega t)$ for a constant $\omega$. Show that there are two values of $\omega$ at which resonance occurs, and find both.

$m = 1$

$k = 4$

$my'' + ky = 0$

$y'' +4y = 0$

$y(t) = c_1\cos (2t) + c_2\sin (2t)$

Now I'm unsure how to show/find the two values for resonance. I know the equation is

$y'' + 4y = 3\cos^3(\omega t)$

but what does that tell me?

Answer 1

Hint: $\cos^3(\omega t) = \cos(3 \omega t)/4 + 3\cos(\omega t)/4$. You get resonance in $m y'' + k y = \cos(r t)$ when $\cos(r t)$ is a solution of the homogeneous equation.