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Thread: Required Transformation...Canonical Form

  1. #1
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    Required Transformation...Canonical Form

    Question: Classify the equation $\displaystyle u_{xx} + 4u_{xy} + 4u_{yy} = 0$, reduce it to standard form, and state the required transformation.

    The textbook says

    $\displaystyle D= b^2 - 4ac = 16-4*4 = 0 \ \Rightarrow \ $ Parabolic

    Then, $\displaystyle \xi = bx - 2ay = 4x - 2y$ and $\displaystyle \eta = y$

    So the equation reduces to $\displaystyle 4u_{\eta \eta} = 0$

    which all makes sense, but I am a little confused with what is the transformation and what is standard form.

    Is [mATH]\xi = bx - 2ay = 4x - 2y[/tex] and $\displaystyle \eta = y$ the required transformation and is $\displaystyle 4u_{\eta \eta} = 0$ standard form?
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  2. #2
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    Yes on both questions (but really only $\displaystyle u_{\eta \eta} = 0$ - divide by 4).
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