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Thread: u_{xxyy}+u_{xx}+u_{yy}=0

  1. #1
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    u_{xxyy}+u_{xx}+u_{yy}=0

    $\displaystyle u_{xxyy}+u_{xx}+u_{yy}=0$

    Separation of variables

    With mixed partial derivatives, is it true that separation of variables wont work?
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  2. #2
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    Quote Originally Posted by dwsmith View Post
    $\displaystyle u_{xxyy}+u_{xx}+u_{yy}=0$

    Separation of variables

    With mixed partial derivatives, is it true that separation of variables wont work?
    Dear dwsmith,

    Let $\displaystyle u(x,y)=f(x).g(x)$

    $\displaystyle u_{xxyy}+u_{xx}+u_{yy}=0$

    $\displaystyle f''(x).g''(y)+f''(x).g(y)+g''(y).f(x)=0$

    $\displaystyle f''(x)\{g''(y)+g(y)\}=-g''(y).f(x)$

    $\displaystyle \dfrac{f''(x)}{f(x)}=-\dfrac{g''(y)}{g''(y)+g(y)}$

    So this partial differential equation is seperable.
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