Cauchy problem for the inhomogeneous wave equation written as $u_{xy}=1$

Question

I have a question on a PDE assignment that's giving me problems interpreting.

Solve the following Cauchy problem for the inhomogeneous wave equation:

$u_{xy} = 1$

$u(x,-x)=6$

$u_x(x,-x)=u_y(x,-x)=12$

That's it. Is $u_{xy} = 1$ the PDE to be solved? Or is it assumed I somehow incorporate it into $u_{tt} - c^2u_{xx}=f(x,t)$?

Any idea would be of great help.

Answer 1

I think $u_{xy}=1$ is the PDE to be solved here. This is the wave equation rewritten in a more convenient system of coordinates. If you look at the derivation of d'Alembert's formula, it is based on switching from the original coordinates to ones where the equation is $u_{xy}=0$. So it's not practical to go back to coordinates where the equation is $u_{tt}-c^2 u_{xx}=0$.

How to solve the given PDE:

1. The homogeneous form $u_{xy}=0$ has general solution $f(x)+g(y)$.
2. Guess a particular solution of nonhomogeneous equation: $xy$.
3. So, look for $u(x,y)= f(x)+g(y)+xy$.
4. Express the initial conditions in terms of $f,g$. They become $f(t)+g(t)-t^2=6$, $f'(t)-t=12$, $g'(t)-t=12$.
5. Integrate to $f(t)=12t+t^2/2+C_1$; $g(t)=12t+t^2/2+C_1$.
6. Pick some $C_1,C_2$ that work.