I have the following equation \begin{equation} F(\theta) + (c)^{\frac{1}{c}-1}\sqrt{\frac{c}{2\pi}}\int\limits_0^\theta \frac{\theta^{\frac{1}{c}} - \tau^{\frac{1}{c}}}{(\theta - \tau)^{3/2}}\exp{\left\{-\frac{(c)^{\frac{1}{c}-1}}{2}\frac{\left[\theta^{\frac{1}{c}} - \tau^{\frac{1}{c}}\right]^2}{\theta - \tau}\right\}}F(\tau)\,d\tau = \\-c k_0\sqrt{\frac{2}{\pi c\theta}}\exp{\left\{-\frac{\left[(c\theta)^{1/c} - x_0\right]^2}{2c\theta}\right\}}. \end{equation} Where $F$ is an unknown function and $c$, $k_0$ and $x_0$ are positive constants.
I am wondering, how to solve this analytically. I know that this can be solved numerically by the iteration method. I do not have much experience with the equations of this sort. Any help or guidance with this will be greatly appreciated, but I really want to understand how to solve it, not just obtain an answer.