Originally Posted by

**shawsend** This is what I'd do: Assuming the mixed partials are equal and you want to integrate with respect to x, than y, then z, I'd write $\displaystyle u_{xyz}(x,y,z) = e^{x} + xy$ as:

$\displaystyle \int \partial_x\left(\frac{\partial^2 u}{\partial z \partial y}\right)=\int (e^x+xy)\partial x$

$\displaystyle \frac{\partial^2 u}{\partial z\partial y}=e^x+x^2/2 y+f(x,y)$

(in general, you can't assume the arbitrary constant is separable so it's f(x,y))

$\displaystyle \int \partial_y\left(\frac{\partial u}{\partial z}\right)=\int\left(e^x+x^2/2y+f(x,y)\right)\partial y$

$\displaystyle \frac{\partial u}{\partial z}=e^x y+\frac{x^2 y^2}{4}+\int f(y,z)\partial y+h(z)$

Then integrating with respect to z:

$\displaystyle u(x,y,z)=e^x y z+\frac{x^2 y^2 z}{4}+\int\int f(y,z)\partial y \partial z+\int h(z)dz$

Also, I'd recommend "Basic Partial Differential Equations" by Bleecker and Csordas.

I think you can drop that last single integral as it's still just an arbitrary function of z such as $\displaystyle f(z)$.