Second order and non-linear differential equation

Question

I'm trying to solve the following differential equation

$\ddot{y}(t)=F\cos(\Omega t)-\frac{1}{m}\left(c-\frac{\alpha^2}{R_l+R_i}\right)\dot{y}(t)-\frac{k}{m}y(t)-\frac{k_3}{m}y^3(t)-g$

with initial condition $y(0)=y_0$. Note that the parameters $F,\ \Omega,\ m,\ c,\ \alpha,\ R_l,\ R_i,\ k, \ k_3,\ g$ and $y_0$ are arbitrary.

I tried using the following code

ode:=diff(y(t),t,t)=F*cos(Omega*t)-(c-alpha^2/(R_l+R_i))*(diff(y(t), t))/m-k*y(t)/m-k3*y(t)^3/m-g;

dsolve({ode, y(0) = y0});

in Maple, but I didn't obtain any solution. Can somebody help me?

Thank you, Ana.

Answer 1

This is similar to the forced Duffing equation $$ \ddot x + \delta\dot x + \alpha x + \beta x^3 = \gamma \cos \omega t \, . $$ No analytical solution is known (it even exhibits chaotic behavior), but approximate solutions can be found.