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

1. ## One more Green's function question...Grrrr

Derive the Green's function for the problem $\displaystyle y'' + y = f(x)$ with $\displaystyle y'(0) = \alpha \ \ y(\pi) = \beta$

2. 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

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