Derive the Green's function for the problem $\displaystyle \displaystyle y'' + y = f(x)$ with $\displaystyle \displaystyle y'(0) = \alpha \ \ y(\pi) = \beta$
To solve the given problem
$\displaystyle (P):\begin{cases} y''+y=f(x)\cr y'(0)=\alpha,y(\pi)=\beta\end{cases}$
you need just solve the initial value problems
$\displaystyle (P_1):\begin{cases} y''+y=0\cr y'(0)=\alpha\end{cases}$ and $\displaystyle (P_2): \begin{cases} y''+y=0\cr y(\pi)=\beta\end{cases}$
If $\displaystyle y_1,y_2$ respectively solve $\displaystyle (P_1),(P_2)$
and their Wronskian is $\displaystyle W=W(t),\ t\in [0,\pi]$,
then the Green's function associated with the problem (P) is
$\displaystyle G(x,t)=\begin{cases} \frac{y_1(x)y_2(t)}{W(t)}, \ x<t \cr \frac{y_1(t)y_2(x)}{W(t)}, \ t<x \end{cases}$
Thus, the solution for $\displaystyle (P)$ reads $\displaystyle u(x)=\int_0^{\pi} G(x,t)f(t)dt$.