differential equation for oscillatory phase The differential equation $$\frac{d^2 x}{dt^2} + k x = 0$$ leads to harmonic solution like $x = \sin(\omega t + \phi)$. Is there a differential equation that leads to a harmonic solution with harmonic phase, like $x = \sin(\omega_a t + \sin(\omega_b t + \phi_b))$?

You can start by taking the first and second derivatives of your solution: $$\frac{d}{dt}\sin(\omega_a t + \sin(\omega_b t + \phi_b))=(\omega_b \cos(\omega_b t+\phi_b)+\omega_a)\cos(\sin(\omega_b t+\phi_b)+\omega_at)\\\\\frac{d^2}{dt^2}\sin(\omega_a t + \sin(\omega_b t + \phi_b))= −(\omega_b \cos(\omega_b t+\phi_b)+\omega_a)^2\sin(\sin(\omega_b t+\phi_b)+\omega_at)−b^2\sin{(\omega_b t+\phi_b)}\cos(\sin(\omega_b t+\phi_b)+\omega_at)$$

Replacing the sin-sin term with $x$ and the cos-sin term with an expression of $dx/dt$, we can now write $$\frac{d^2 x}{dt^2}=−(\omega_b \cos(\omega_b t+\phi_b)+\omega_a)^2 x−\frac{\omega_b^2\sin{(\omega_b t+\phi_b)}}{\omega_b \cos(\omega_b t+\phi_b)+\omega_a}\frac{dx}{dt}$$

You might be able to simplify it, but this is an example of the differential equation that you were looking for