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Partial diff eqn problem

  1. Oct 27, 2009 #1
    1. The problem statement, all variables and given/known data
    the substitution x=es, y=et converts f(x,y) into g(s,t) where g(s,t)=f(es, et). If f is known to satisfy the partial differential equation

    x2(d2f/dx2) + y2(d2f/dy2) + x(df/dx) + y(df/dy) = 0

    show that g satisfies the partial-differential equation

    (d2g/ds2) + (d2g/dt2) = 0

    3. The attempt at a solution

    I feel like this is a simple problem - All I need to figure out is how to find the partial derivative of f(x,y) and the double partial derivative, but I'm not sure how to do it. Are the values of x and y relevant for the first part of the question?

    what would df/dx be? D1f(x,y) = (y)f'(x,y) by the chain rule? and similarly df/dy = D2f(x,y) = (x)f'(x,y)? Someone please help!
     
  2. jcsd
  3. Oct 28, 2009 #2

    lanedance

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    Homework Helper

    i assume they're meant to be partials... & I dont really understand what you've written at the end of the post but here's some ideas I hope help

    now as you're given
    [tex] x(s)=e^s[/tex]
    [tex] y(t)=e^t[/tex]

    invert these and then re-write as
    [tex] s(x)=ln(s)[/tex]
    [tex] t(y)=ln(y)[/tex]

    so the relation function g & f is given as
    [tex] g(s,t) = f(x(s), y(t)) [/tex]

    we can re-write it and think of it as
    [tex] f(x,y) = g(s(x), t(y)) [/tex]

    now try taking a partial w.r.t x and using the chain rule on the RHS

    [tex] \frac{\partial f(x,y)}{\pratial x} = \frac{\partial }{\partial x} g(s(x), t(y)) [/tex]
    the partial w.r.t. x means the other variable (y) in this case is kept constant
     
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