We consider a forced and damped harmonic oscillator, described by \begin{align} \ddot{x}(t) + 2 \kappa \dot{x}(t) + \omega_0^2x(t) = F(t) \, \, \, \, \, \, \,\, \, \, \, \, \, \,\, \, \, \, \text{(1)} \end{align} Here, $\omega_0$ and $\kappa$ are real constants with $0 \leq \kappa < \omega_0$, and $F: I \rightarrow \mathbb{C}$ is a continuous function.
Find a complex solution $z_p$ of the differential equation $(1)$ using the method of variation of parameters, with a fundamental complex system $\phi_1, \phi _2$ of the homogeneous equation.
I tried to solve the fundamental system and I found the solutions $\phi_1(t) = e^{-k⋅t+i \cdot t \cdot \sqrt{\omega_0^2 - \kappa^2}}$ and $\phi_2(t) =e^{-k⋅t-i \cdot t \cdot \sqrt{\omega_0^2 - \kappa^2}}$. $\, $Now I have to solve the following system to find $c_1$ and $c_2$ \begin{align} \begin{cases} \dot{c_1} \phi_1 + \dot{c_2}\phi_2 =0 \\ \dot{c_1}\dot{\phi_1} + \dot{c_2}\dot{\phi_2} = F(t) \end{cases} \end{align} but I have some to trouble to solve it.
Any suggestions? Thanks in advance!