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Method of characteristics for a 1st order quasi-linear PDE.

  1. Mar 27, 2008 #1

    I'm looking over the examples in my book for this problem and the general approach is

    a(x,y,z)*u_x + b(x,y,z)*u_y = c(x,y,z)

    where u(x,y)

    I have the following problem in my notes:

    1/x * u_x + 1/y * u_y = x^2 * sqrt(z)

    and I get the solution easily because of the format:

    Another problem i did was

    y*u_x - x * u_y = 2xyu

    where i multiplied both sides by 1/xy to get:

    1/x*u_x - 1/y * u_y = 2u

    the solution followed easily because of the classic format:

    however, what i don't get is this format

    u*u_x + y*u_y = x

    the problem is an ivp with the following characteristics:

    x = s, y = s, u = 2s

    z = u(x_o,y_o) = u(s,s) = 2s

    how do i get the solutin when it's not in the standard format? Or am i misinterpreting

    a(x,y,z)*u_x + b(x,y,z)*u_y = c(x,y,z)

    in the sense that the coefficient functions (a(x,y,z), b(x,y,z), c(x.y,z) can be any variable x,y,u(x_o,y_o)?

    I don't have the book and all my examples are written in the convenient form i'm after which confuses me

    thank you
  2. jcsd
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