Okay, so I end up with something along the lines of...
K [x, 2y, 3z] = - [dU/dx, dU/dy, dU/dz]
and then we integrate
U = -k [ (1/2)x2, y2,(3/2)z2]
Would it be proper to add a constant at the end of each part, noting that it's terms of the other two variables? Something like
U = -k [ (1/2)x2...