# Method of Characteristics

1. Dec 14, 2015

### Sturk200

1. The problem statement, all variables and given/known data
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.

3. 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!

2. Dec 16, 2015