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Math Help - u_{xx}+u_{yy}=0

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

    Separation of Variables

    u_{xx}+u_{yy}=0

    \displaystyle e^{rx+sy}=e^{rx}e^{sy}

    u(x,y)=\varphi(x)\psi(y)

    \varphi''(x)\psi(y)+\varphi(x)\psi''(y)=0


    \displaystyle\frac{\varphi''(x)\psi(y)+\varphi(x)\  psi''(y)}{\varphi(x)\psi(y)}=0

    \displaystyle \frac{\varphi''(x)}{\varphi(x)}+\frac{\psi''(y)}{\  psi(y)}=0

    \displaystyle \frac{\varphi''(x)}{\varphi(x)}=-\frac{\psi''(y)}{\psi(y)}

    \displaystyle \frac{\varphi''(x)}{\varphi(x)}=\lambda \ \mbox{and} \ \frac{\psi''(y)}{\psi(y)}=-\lambda

    \Rightarrow \varphi''(x)-\lambda\varphi(x)=0\Rightarrow \mbox{sol.s} \ \cos{(x\sqrt{\lambda})} \ \mbox{and} \ \sin{(x\sqrt{\lambda})}

    \Rightarrow \psi''(y)+\lambda\psi(y)=0\Rightarrow \mbox{sol.s} \ e^{\pm y\sqrt{\lambda}}

    How are those solutions obtained?

    Also, I can't tell if the books has e^{\pm y\sqrt{\lambda}} or e^{\pm y^{\sqrt{\lambda}}}.
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  2. #2
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    Quote Originally Posted by dwsmith View Post
    Separation of Variables

    u_{xx}+u_{yy}=0

    \displaystyle e^{rx+sy}=e^{rx}e^{sy}

    u(x,y)=\varphi(x)\psi(y)

    \varphi''(x)\psi(y)+\varphi(x)\psi''(y)=0


    \displaystyle\frac{\varphi''(x)\psi(y)+\varphi(x)\  psi''(y)}{\varphi(x)\psi(y)}=0

    \displaystyle \frac{\varphi''(x)}{\varphi(x)}+\frac{\psi''(y)}{\  psi(y)}=0

    \displaystyle \frac{\varphi''(x)}{\varphi(x)}=-\frac{\psi''(y)}{\psi(y)}

    \displaystyle \frac{\varphi''(x)}{\varphi(x)}=\lambda \ \mbox{and} \ \frac{\psi''(y)}{\psi(y)}=-\lambda

    \Rightarrow \varphi''(x)-\lambda\varphi(x)=0\Rightarrow \mbox{sol.s} \ \cos{(x\sqrt{\lambda})} \ \mbox{and} \ \sin{(x\sqrt{\lambda})}

    \Rightarrow \psi''(y)+\lambda\psi(y)=0\Rightarrow \mbox{sol.s} \ e^{\pm y\sqrt{\lambda}}

    How are those solutions obtained?

    Also, I can't tell if the books has e^{\pm y\sqrt{\lambda}} or e^{\pm y^{\sqrt{\lambda}}}.
    The two final DEs are second-order linear constant coefficient homogeneous. I suggest you read Differential Equations Tutorial to refresh your memory how to solve DEs of this type.
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  3. #3
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    Quote Originally Posted by Prove It View Post
    The two final DEs are second-order linear constant coefficient homogeneous.
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