Nonhomogeneous Equations & Resonance · Hard problem

For $y''+2y'+5y=\cos(\omega t)$ with $\omega>0$, the steady-state response lags the forcing by a phase angle $\varphi$ with $\tan\varphi=\dfrac{2\omega}{5-\omega^2}$. Find $\omega$ for which the phase lag is exactly $\dfrac\pi4$.