I solve Duffing equation with Van Der Paul's method:
\begin{align} \ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t) \end{align}
with $$x(t) = Re[A(t) \exp(i \omega_0 t)]$$ and $$A(t) = a(t) \exp(i \phi(t)),$$ where $a(t), \phi(t)$ are slowly changing variables. So, I have got this system of differential equations:
\begin{align} \dot{a} + \gamma a = \dfrac{f}{2 \omega_0} \sin(\Psi), \\ \dot{\Psi} = \delta + \dfrac{3 \beta a^2}{8 \omega_0} - \dfrac{f}{2 \omega_0 a} \cos(\Psi), \end{align}
where $\gamma, \delta = \Omega - \omega_0 \ll 1, \beta, \omega_0$ are parameters. Can I solve this somehow? I need to get equation of motion $x(t)$ with defined initial conditions.
Without 'force' it's easy to integrate the system and find $a(t), \Psi(t)$. But I need to take 'force' into account. And I can square both equations to get rid of sine and cosine.