where u_x is u(x,y) differentiated with respect to x and u_xy is u(x,y) twice differentiated with respect to x and with respect to y
How would I go about solving this?
Thanks for any help.
What you present is just one solution. Here are two more
(1) $\displaystyle u = e^{-2y} \sin x $
(2) $\displaystyle u = x e^{-2y} + y^2 $
Why not let $\displaystyle v = u_x$ and see where that gets you.
Since only differentiation with respect to y is involved, you can solve that as the ODE
$\displaystyle \frac{dv}{dy}= -2y$. Of course, the "constant" may be a function of x. What does that give you for v? What does that make the orginal equation?
Remember that the general solution to a partial differential equation may involve unknown functions of the variables rather than unknown constants.