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Math Help - A PDE transformation

  1. #1
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    A PDE transformation

    Hi there,
    In the notes I'm reading, they have stated the following PDE:
    u_{xx}+u_{yy}+2au_x+2bu_y+cu=d.
    Then, by defining:
    v(x,y)=e^{ax+by}u(x,y)
    The PDE becomes, in terms of v:
    v_{xx}+v_{yy}+(c-(a^2+b^2))v=e^{ax+by}d
    I don't see how they made this transformation. Should I taking partial derivatives of u as a function of v?

    Thanks for your time.
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  2. #2
    A Plied Mathematician
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    Should I taking partial derivatives of u as a function of v?
    I would write u as a function of v and the exponential function there. That is,

    u(x,y)=e^{-ax-by}\,v(x,y).

    Then start taking the partial derivatives you need to assemble the ingredients of your first equation. Then plug everything in and simplify.
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  3. #3
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    Cheers.
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  4. #4
    A Plied Mathematician
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    Did you get your answer?
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  5. #5
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    Yeah, took a bit, but I got it out. Thanks for your advice.
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  6. #6
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    You're welcome. Have a good one!
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