I am trying to find a general form solution of the following integral equations \begin{align} \int_0^\infty q F(q) J_1 (qr) \, \mathrm{d} q = 0 \, , & \text{ for } r \ge R \, , \\[6pt] \int_0^\infty q (1-i\alpha q) F(q) J_1 (qr) \, \mathrm{d} q = -\frac{ir(4h^2-r^2)}{(h^2+r^2)^{7/2}} \, , & \text{ for } r \le R \, , \end{align} wherein $J_1$ is the Bessel functions of the first kind of order 1, $i$ being the imaginary unit and $\alpha$ and $h$ are known lengths. The goal is the determination of the unknown complex valued function $F(q)$.
I was wondering if there exists a commonly used technique for the resolution of such homogenous integral equations. Any help would be highly appreciated. Thank you.
R