You might try looking up the term in your textbook's index (or, if you don't have a required text, ask your professor where to find a definition of that term).

As for the solution, like I said, go for a separable solution. That is, assume that

\(\displaystyle u(x,y)=X(x)Y(y)\).

Plug this into your system, and you should be able to get all the x's to one side, and the y's to the other, in both original equations. From that, you can deduce that each of the sides is constant. Use that to convert into 4 ODE's.