How do I solve the following equation for x:
$$\ddot{x} + 2\beta x + \alpha^2 x= F \cos[\omega(t)t + \phi]$$
where $\omega(t)$ is a linear function given by:
$$\omega(t) = mt + \omega_c$$
I have tried expressing the RHS of the second ODE in it's fourier modes:
$$\cos[\omega(t)t + \phi] = \mathcal{F}^{-1}\{\mathcal{F}\{\cos[\omega(t)t + \phi]\}\}\\ =\frac{1}{2\pi}\int^\infty_{-\infty}\mathcal{F}\{\cos[\omega(t)t + \phi]\}\,e^{i\omega't}\,d\omega'$$
and then proceed with solving the equation for each individual fourier mode:
$$\ddot{x} + 2\beta x + \alpha^2 x= F\, \mathcal{F}\{\cos[\omega(t)t + \phi]\}$$
For the resultant solution $x_\omega(t)$, the final particular solution will thus be:
$$x_p(t) = \frac{1}{2\pi}\int^\infty_{-\infty}x_\omega(t) \, e^{i\omega't}\, d\omega'$$
However, I am unable to perform the fourier transformation in the third line analytically thus far. Is there a better way to solve this second ODE?
I am not sure but maybe it would help expressing the un-known function as a complex one: $z(t)=x(t) + iy(t)$ and then searching for a solution of the form $z(t)=z_0 e^{i \omega t}$. At least thats how we solved it in my mechanics class (where the frequency was constant). Also for periodic forces you can Fourier expand everything and then play only with the coefficients, otherwise if you want to use the Fourier transform you will probably end up in something like this:
$$z(t) = \int _{-\infty} ^{+\infty} \frac{f(\omega) e^{i\omega t} d\omega}{a^2 - \omega^2 + 2i\beta \omega}$$
Where $f(\omega) = F cos( \omega(t) t + \phi)$ (maybe theres also a mass in there). Now this integral has two poles:
$$\omega_{\pm} = \frac{2i\beta \pm \sqrt{4a^2 - 4\beta ^2}}{2}$$
and using the Residue theorem you can calculate it, thus finding $x(t)=Re\{z(t)\}$ As i said i am unsure cause i haven't seen any problem with time-dependent frequency, i am just sketching the general solution so you can maybe take something from it.