Find the general solution of the following PDE.

u_{xxyy}= 0

I know I have to go from right to left i.e integrate with respect to y, then y then x and x.

So here it goes

u_{xxy = }F(x) Where F(x) is a constant in terms of x.

u_{xx =}yF(x) +F_{1}(x) Where F_{1}(x) is a constant in terms of x.

u_{x = }yxF(x) + xF_{1}(x) + F_{2}(y) Where F_{2}(y) is a constant in terms of y.

u = u(x,y) = 1/2 (yx^{2}F(x) ) + 1/2 (x^{2}F_{1}(x) ) + xF_{2}(y) + F_{3}(y) Where F_{3}(y) is a constant in terms of y.

Is that correct?

Thanks