Chapter 11.6, Problem 21E

Derive a formula using a Green’s function for the solution to the following nonhomogeneous equation with nonhomogeneous boundary conditions

$y^{″}\left(x\right)=-f\left(x\right);y\left(0\right)=\alpha ,y\left(1\right)=\beta$.

[Hint: Let $y=u+v$, where $u^{″}=0$ and $u$ satisfies the nonhomogeneous boundary conditions in $\left(25\right)$, while $v^{″}=-f$ and $v$ satisfies the corresponding homogeneous boundary conditions where $\alpha =\beta =0$.]