Consider the following ODE:
$\ddot{\phi} + 2c\dot{\phi} + (1 + \epsilon \cos 2\Omega t)\phi = 0$,
where $\epsilon, c << 1$ and $\Omega \approx 1$.
(4.1) Seek a solution in the form
$\phi = B(t)\cos\Omega t + D(t)\sin\Omega t$.
(4.2) Upon substitution of (2) into (1), omit small terms involving $\ddot{B}$, $\ddot{D}$, $c\dot{B}$, and $c\dot{D}$.
(4.3) Omit the non-resonant terms, i.e. terms involving $\cos 3\Omega t$ and $\sin 3\Omega t$.
(4.4) Collect like terms and obtain a set of equations for $B(t)$ and $D(t)$.
(4.5) Reduce the set obtained in Part (4.4) to a single equation for $D(t)$ and find its general solution.
(4.6) On the basis of the solution obtained in Part (4.5), determine the range of $\Omega$ for which
parametric resonance occurs in the system.
