Method of Characteristics for Solving Partial Differential Equations

[Sturk200]

Homework Statement

I am trying to solve the following equation using the method of characteristics:

∂u/∂x + (xy)(∂u/∂y) + 2x²zLog[y](∂u/∂z) = 0

I'm really just trying to follow along the solution provided here:
http://www.ucl.ac.uk/~ucahhwi/LTCC/sectionA-firstorder.pdf

on page 9.

The Attempt at a Solution

I get that we have to parameterize the dependent variables so x=x(r), y=y(r), z=z(r). Then demand that du/dr=0, from which we get a system of three autonomous equations:

dx/dr = 1; dy/dr = xy; dz/dr = 2x²Log[y].

Then we have to solve these to get the form of u. My trouble is with solving the z equation. According to the handout I linked to above the right hand side of the dz/dr equation becomes:

2x²zLog[y] = -r⁴zLog[y₀], where r=x and y=y₀Exp[r^2/2] (these expressions for x and y come from solving the other two equations).

I don't understand how we can make this step. When I plug in for y and x to the left side of the above I come up with y=2zr^2Log[y₀] + r^4/2 ...

Can somebody tell me what I'm doing wrong and how to get the answer given in the handout?

Thank you!

 

[haruspex]

I come up with y=2zr²Log[y₀] + r⁴/2...

I don't see how you get that. Please post your working.