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Math Help - partial differential equation

  1. #1
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    partial differential equation

    How to solve the attached coupled equations for \psi and 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 \dfrac{\partial^2 \psi}{\partial x^2}+\dfrac{\partial^2 \psi}{\partial y^2}=k^2\psi.

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

    \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 \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 C is a constant.

    I leave the rest to you.
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