Oi. That's a de-coupled system (at least, the first equation is independent of y). The solution to the first equation is quite nasty: it involves Airy functions and gamma functions. Or, alternatively, you could write the solution in terms of Bessel functions. Here's WolframAlpha's solution.

Once you have that, you can plug that into your other DE and try to solve that. (Good luck.)

My recommendation: solve it numerically.