It's well known that the solution of the differential equation: $\ddot x(t)+\omega^2x(t)=\sin(\psi t)$ has the form: $x(t)=C_1 \sin(\omega t)+C_2 \cos(\omega t)-\frac{\sin(\psi t)}{\psi^2-\omega^2}$ Obviously, if $\psi=\omega$, there is a resonance and the amplitude of the oscillations diverges.
My question is: what happens if $\psi$ is a normal distribuited random variable with mean value $\omega$ and variance $\sigma$? Thanks in advance.