I am studying an equation that looks like $x'' + \omega^2 x = 0$, where $\omega^2$ is discrete and only takes on values like $C m(m + 2)$ where $C$ is a constant and $m = 1, 2, 3, \dots$
I'm pretty confused regarding whether the solutions would be different from the case where $\omega^2$ is continuous.
Any help would be appreciated, thank you.
Assuming that $C>0$, then the usual trigonometric solutions work with angular frequency $$ \omega = \sqrt{Cm(m+2)}, $$ so the general solution $m \geq 1$ looks like $$ x(t) = c_1 \sin \omega t + c_2 \cos \omega t, $$ where $c_1$ and $c_2$ are arbitrary constants that depend on initial/boundary values. You can view this family of sinusoids here.