I'm trying to find an analytical solution for the following differential equation that contains an integral term:
\begin{align} \frac{d{v}_p(t)}{dt} &= \alpha + \beta\frac{d{v}_f(t)}{dt} + \gamma\frac{d}{dt}\big({v}_p(t)-{v}_f(t)\big) + \label{eq:force_eq} \\ &+ \mu \big( {v}_p(t)-{v}_f(t) \big)^2 + \nu \int_{0}^{t}\frac{\frac{d}{d\tau}({v}_p(\tau)-{v}_f(\tau))}{\sqrt{(t-\tau)}} d\tau \nonumber \end{align} Where $\alpha, \beta, \gamma, \mu$ and $\nu$ are constant and the funcions $v_f(t)$ and $dv_f(t)/dt$ are known $\big($you might assume $v_f(t) = v_{0} cos(\omega t)\big)$. I want to solve this equation for $v_p(t)$. Anyone can help me with this?