Solving the PDE u_(xy) = ku with some initial conditions

[quasar987]

Homework Statement

Does anyone know how to solve this PDE for u:R-->R and some initial conditions?

$$u_{xy}=ku$$

where k is a positive constant.

Or this one, also for u:R-->R and some initial conditions:

$$u_{tt}=u_{xx}-Ku$$

where K is a positive constant.

The Attempt at a Solution

I can solve the second one for u:[0,L]-->R and the boundary conditions of the fixed string u(0,t)=u(L,t)=0 by separation of variable. The solution, a superposition of normal modes, differs only from the solution of the usual wave equation u_tt=u_xx in that the frequencies are $$\sqrt{\lambda_n^2+K}$$, where lambda_n=npi/L is the usual nth eigenfrequency for the usual wave equation.

In the usual wave equation solution, the normal modes are superpositions of traveling waves. And traveling waves are the general solution to the "free" wave equation u_tt=u_xx for u:R-->R, obtained by d'Alembert's method. Can I conclude that the solution for u:R-->R of u_{tt}=u_{xx}-Ku are also traveling waves?

 

[Rainbow Child]

The first one reduces to the second one by the change of variables
$$x=\frac{1}{2}\,(v+u),\,y=\frac{1}{2}\,(v-u)$$