I would like to find the solution to the following dual integral equations $$ \int_0^\infty (1+\alpha t) A(t) J_1 (\rho t) \, \mathrm{d} t = f(\rho) \quad \quad (0 \le \rho < d) \, , \\ \int_0^\infty A(t) J_1 (\rho t) \, \mathrm{d} t = 0 \quad\quad (\rho>d) \, , $$ where $\alpha$, $d$ and $f(\rho)$ are known quantities and $A(t)$ is the unknown function. The system with $\alpha=0$ has originally been considered by Titchmarsh (Theory of Fourier Integrals, Oxford, 1937) p. 337
I was wondering if someone here has already come across a similar system before. Any help would be highly appreciated.
Thanks
R