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Thread: partial differential equation

  1. #1
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    partial differential equation

    How to solve the attached coupled equations for $\displaystyle \psi$ and $\displaystyle u$. Please help me. Its very complex.
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  2. #2
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    Re: partial differential equation

    You can try separation of variables.

    The given equation is $\displaystyle \dfrac{\partial^2 \psi}{\partial x^2}+\dfrac{\partial^2 \psi}{\partial y^2}=k^2\psi$.

    Let $\displaystyle \psi(x,y)=a(x)b(y)$ then $\displaystyle \dfrac{\mathrm{d}^2 a}{\mathrm{d} x^2}b+a\dfrac{\mathrm{d}^2 b}{\mathrm{d} y^2}=k^2ab$

    $\displaystyle \dfrac{1}{a}\dfrac{\mathrm{d}^2 a}{\mathrm{d}x^2}+\dfrac{1}{b}\dfrac{\mathrm{d}^2 b}{\mathrm{d}y^2}=k^2$

    This means $\displaystyle \dfrac{1}{a}\dfrac{\mathrm{d}^2 a}{\mathrm{d}x^2}=k^2-\dfrac{1}{b}\dfrac{\mathrm{d}^2 b}{\mathrm{d}y^2}=C$ where $\displaystyle C$ is a constant.

    I leave the rest to you.
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