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Math Help - reduce PDE to normal form

  1. #1
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    reduce PDE to normal form

    Hi, I am having difficulty doing these 2 questions. I tried the change of coordinates but that got very messy and got me nowhere.

    Reduce to canonical form:
    1. y^2u_{xx}+2xyu_{xy}+x^2u_{yy}=0
    2. u_{xx}-2xyu_{xy}=0, (x \ne 0)

    Thanks for your help.
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  2. #2
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    Re: reduce PDE to normal form

    For the first one, since B^2 - 4AC = 0, then the PDE is parabolic and the normal or standard form is u_{ss} + l.o.t.s. = 0. Substitute the change of variables in and collect about the like higher or terms (i.e. u_{rr}, u_{rs}, u_{ss}). Setting the coefficient of u_{rr} and u_{rs} = 0 gives

    x r_x + yr_y = 0.

    This we solve giving r = R(y/x). Now R can be as anything involving y/x and s can be chosen as really anything provided that r_xs_y - r_ys_x \ne 0. For example,

    r = y/x, s = y

    or

    r = \ln x - \ln y, s = \ln y.

    Both will lead to your standard form.
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