Ideas on solving a 4th order partial differential equation I am working on solving a 4th order PDE of the form
$$u_{xxxx}+2u_{xxyy}+u_{yyyy}=0$$
I was thinking that a seperation of variables might work, but that does not seem to be the case for $u = XY$, I get 
$$X^{(4)}Y+2X^{(2)}Y^{(2)}+XY^{(4)}=0$$
THis seems to be dead end as I cannot just set it equal to a constant no more. I am hoping maybe there is a cleaver substation or something that will allow me to continue with separation of variables, but so far I have been unsuccessful with that.
 A: Let $\begin{cases}p=x+iy\\q=x-iy\end{cases}$ ,
Then $u_x=u_pp_x+u_qq_x=u_p+u_q$
$u_{xx}=(u_p+u_q)_x=(u_p+u_q)_pp_x+(u_p+u_q)_qq_x=u_{pp}+u_{pq}+u_{pq}+u_{qq}=u_{pp}+2u_{pq}+u_{qq}$
$u_{xxx}=(u_{pp}+2u_{pq}+u_{qq})_x=(u_{pp}+2u_{pq}+u_{qq})_pp_x+(u_{pp}+2u_{pq}+u_{qq})_qq_x=u_{ppp}+2u_{ppq}+u_{pqq}+u_{ppq}+2u_{pqq}+u_{qqq}=u_{ppp}+3u_{ppq}+3u_{pqq}+u_{qqq}$
$u_{xxxx}=(u_{ppp}+3u_{ppq}+3u_{pqq}+u_{qqq})_x=(u_{ppp}+3u_{ppq}+3u_{pqq}+u_{qqq})_pp_x+(u_{ppp}+3u_{ppq}+3u_{pqq}+u_{qqq})_qq_x=u_{pppp}+3u_{pppq}+3u_{ppqq}+u_{pqqq}+u_{pppq}+3u_{ppqq}+3u_{pqqq}+u_{qqqq}=u_{pppp}+4u_{pppq}+6u_{ppqq}+4u_{pqqq}+u_{qqqq}$
$u_{xxy}=(u_{pp}+2u_{pq}+u_{qq})_y=(u_{pp}+2u_{pq}+u_{qq})_pp_y+(u_{pp}+2u_{pq}+u_{qq})_qq_y=iu_{ppp}+2iu_{ppq}+iu_{pqq}-iu_{ppq}-2iu_{pqq}-iu_{qqq}=iu_{ppp}+iu_{ppq}-iu_{pqq}-iu_{qqq}$
$u_{xxyy}=(iu_{ppp}+iu_{ppq}-iu_{pqq}-iu_{qqq})_y=(iu_{ppp}+iu_{ppq}-iu_{pqq}-iu_{qqq})_pp_y+(iu_{ppp}+iu_{ppq}-iu_{pqq}-iu_{qqq})_qq_y=-u_{pppp}-u_{pppq}+u_{ppqq}+u_{pqqq}+u_{pppq}+u_{ppqq}-u_{pqqq}-u_{qqqq}=-u_{pppp}+2u_{ppqq}-u_{qqqq}$
$u_y=u_pp_y+u_qq_y=iu_p-iu_q$
$u_{yy}=(iu_p-iu_q)_y=(iu_p-iu_q)_pp_y+(iu_p-iu_q)_qq_y=-u_{pp}+u_{pq}+u_{pq}-u_{qq}=-u_{pp}+2u_{pq}-u_{qq}$
$u_{yyy}=(-u_{pp}+2u_{pq}-u_{qq})_y=(-u_{pp}+2u_{pq}-u_{qq})_pp_y+(-u_{pp}+2u_{pq}-u_{qq})_qq_y=-iu_{ppp}+2iu_{ppq}-iu_{pqq}+iu_{ppq}-2iu_{pqq}+iu_{qqq}=-iu_{ppp}+3iu_{ppq}-3iu_{pqq}+iu_{qqq}$
$u_{yyyy}=(-iu_{ppp}+3iu_{ppq}-3iu_{pqq}+iu_{qqq})_y=(-iu_{ppp}+3iu_{ppq}-3iu_{pqq}+iu_{qqq})_pp_y+(-iu_{ppp}+3iu_{ppq}-3iu_{pqq}+iu_{qqq})_qq_y=u_{pppp}-3u_{pppq}+3u_{ppqq}-u_{pqqq}-u_{pppq}+3u_{ppqq}-3u_{pqqq}+u_{qqqq}=u_{pppp}-4u_{pppq}+6u_{ppqq}-4u_{pqqq}+u_{qqqq}$
$\therefore u_{pppp}+4u_{pppq}+6u_{ppqq}+4u_{pqqq}+u_{qqqq}-2u_{pppp}+4u_{ppqq}-2u_{qqqq}+u_{pppp}-4u_{pppq}+6u_{ppqq}-4u_{pqqq}+u_{qqqq}=0$
$16u_{ppqq}=0$
$u_{ppqq}=0$
$u(p,q)=qE(p)+F(p)+pG(q)+H(q)$
$u(x,y)=(x-iy)E(x+iy)+F(x+iy)+(x+iy)G(x-iy)+H(x-iy)$
