In solving a specific problem in fluid mechanics, I have obtained a solution to a problem resembling the following. I won't provide the details of the PDEs in exact form, but I will say that I was solving two linear second-order PDEs, which are coupled and have a variety of boundary conditions. Having used a Fourier sine transform to solve the problem, I have obtained a solution of the form
$$ f(x,y)=\dfrac{2}{\pi}\int_{0}^{\infty}\int_{0}^{x}\left(\dfrac{\sinh\omega^{2}x}{\sinh\omega^{2}a}-\dfrac{1}{\omega^{3}}\right)f(\xi,0)\cosh\omega^{2}(x-\xi)\sin\omega y{\rm d}\xi{\rm d}\omega,\qquad\text{ in }0\le x\le a,y\ge 0. $$
I am completely unfamiliar with integral equations, other than a very faint awareness of what they are. Does the above look like it comes from a specific family of integral equations? If so, what is it called, and what is a good starting point on where I can look if I want to obtain a closed-form solution (if indeed it is possible)?
EDIT: Integral originally contained $f(x,0)$ - it should be $f(\xi,0)$, as seen above.