Second-order, linear, homogeneous, hyperbolic PDE. Solvable?

[PDE_warrior]

First, my deepest apologies if I am asking a trivial question, or asking it in the wrong forum.

I am trying to solve a PDE, which I have already reduced to canonical form and simplified to the full extent of my abilities. The PDE is:
u_xy + a(x,y) u_x + b u_y = 0, with a(x,y)=2/(x+y) and b=-1/2.

The method of separation of variables does not work. I have tried posing u(x,y)=v(x,y) w(x,y) and choosing v in such a way as to simplify the resulting equation, to no avail.

Since I know there are two families of solutions involving each an arbitrary function of one variable, I tried setting u(x,y)=v(x,y,F(z(x,y))), with F arbitrary. This leads to two cases: z_x = 0 OR z_y = 0, as well as an ODE that must be satisfied to remove the dependency in F_z. I can solve that ODE for either case, but either way the leftover equation then can only be satisfied by setting F=0.

I am aiming for a solution of the form u(x,y) = anything involving x, y, up to two arbitrary functions and any number of integrations, so that I could potentially plug in the proper functions once I have decided exactly what the boundary conditions should be. In particular, any solution expressed using a function that solves the adjoint problem (as is found in Garabedian) is totally useless to me since the adjoint problem is just as hard to solve as the original one.

I am thankful for any and all help anyone may provide, even just suggestions or pointing me in the correct direction.

 

[jvicens]

Thank you for reaching out for help with your PDE problem. It seems that you have already put in a lot of effort in trying to solve it using various methods. However, it is important to note that not all PDEs can be solved using separation of variables or other analytical techniques. Some PDEs require numerical methods for their solution.

In your case, it might be helpful to try using numerical methods such as finite difference or finite element methods. These methods involve discretizing the domain and approximating the solution at discrete points. They can handle complex PDEs that cannot be solved analytically.

Another approach you can try is to use a computer algebra system such as Mathematica or Maple to solve your PDE. These programs have built-in functions for solving PDEs and can handle many different types of equations.

I understand that you are looking for a solution in a specific form, but it is important to keep in mind that the solution of a PDE is not always expressible in a closed form. Sometimes, the solution may involve special functions or infinite series. It might be helpful to consult with a mathematician or a numerical analyst for further guidance on your specific problem.

I hope this helps and wish you all the best in finding a solution to your PDE. If you have any further questions, please do not hesitate to ask. Keep persevering and I am sure you will find a solution eventually.