I can't figure this out. The equation is $\displaystyle u_{xx}+4u_{yy}=0$. Solving for y', I get y1 = 2ix + c1 and y2 = -2ix + c2.

Then v = y1 - 2ix and w = y2 + 2ix.

I keep getting $\displaystyle -4u_{vv}+8u_{vw}-4u_{ww}=0$ and $\displaystyle u_{vv}+2u_{vw}+u_{ww}=0$ from the second partials with respect to x twice and y twice, respectively. This doesn't tell me anything, because these two equations are equivalent. Please advise.