The last term vanishes because $z$ is unaffected by $x$? I'm not entirely convinced that's necessarily the case, but it's a bit late where I am and I need some sleep. I'll return to the problem tomorrow and try to solve it: I think your hint has more or less cleared the issue for me.
Out of...
Homework Statement
Consider heat flow in a long circular cylinder where the temperature depends only on t and on the distance r to the axis of the cylinder. Here r=\sqrt{x^2+y^2} is the cylindrical coordinate. From the three-dimensional heat equation derive the equation U_t=k(U_{rr}+2U_r/r)...