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Method of Characteristics

  1. Dec 14, 2015 #1
    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. jcsd
  3. Dec 16, 2015 #2

    haruspex

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