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Math Help - One more Green's function question...Grrrr

  1. #1
    Junior Member BrooketheChook's Avatar
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    One more Green's function question...Grrrr

    Derive the Green's function for the problem  \displaystyle y'' + y = f(x) with  \displaystyle y'(0) = \alpha \ \ y(\pi) = \beta
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  2. #2
    Super Member Rebesques's Avatar
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    To solve the given problem

    (P):\begin{cases} y''+y=f(x)\cr y'(0)=\alpha,y(\pi)=\beta\end{cases}

    you need just solve the initial value problems
    (P_1):\begin{cases} y''+y=0\cr y'(0)=\alpha\end{cases} and (P_2): \begin{cases} y''+y=0\cr y(\pi)=\beta\end{cases}

    If y_1,y_2 respectively solve (P_1),(P_2)
    and their Wronskian is W=W(t),\ t\in [0,\pi],
    then the Green's function associated with the problem (P) is

    G(x,t)=\begin{cases} \frac{y_1(x)y_2(t)}{W(t)}, \ x<t \cr \frac{y_1(t)y_2(x)}{W(t)}, \ t<x \end{cases}

    Thus, the solution for (P) reads u(x)=\int_0^{\pi} G(x,t)f(t)dt.
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