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Math Help - Quasi linear PDE help

  1. #1
    Newbie mukmar's Avatar
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    Quasi linear PDE help

    I'm trying to understand a solved problem in our notes and can't find a similar technique used elsewhere, so I'm asking for some assistance in understanding what exactly they did.

    The problem:
    Find a general solution to the equation  x^2u_x + y^2u_y = (x + y)u
    and the particular solution when u(x,1) = 1

    Solution:
    The characteristic equations are:
    \frac{dx}{dt} = x^2, \frac{dy}{dt} = y^2, \frac{du}{dt} = (x + y)u

    I don't really understand these next steps they do which is:

    These imply that for some constant \alpha
    \frac{x' - y'}{x - y} = \frac{u'}{u}, and \frac{1}{x} - \frac{1}{y} = \alpha

    from which it follows that two integrals are for constants ( \beta, \gamma)
    x - y = \beta u, or u = \gamma xy

    There are a few more steps after this, but I guess I'll ask about those if I don't understand them after I understand these steps.

    Thanks for any assistance.
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  2. #2
    Super Member Rebesques's Avatar
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    From the first two, x'-y'=x^2-y^2=(x-y)(x+y) and using the third, x'-y'=(x-y)(u'/u)... Integrate by separating variables after that.
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