= Solution
For a fixed source point $\xi\in(a,b)$, the <Green function> is defined by
$$
\boxed{L_xG(x;\xi)=\delta(x-\xi),\qquad G(a;\xi)=G(b;\xi)=0,}
$$
where $\delta$ is the <Dirac delta distribution> and the operator acts on $x$. The corresponding solution is
$$
\boxed{y(x)=\int_a^bG(x;\xi)f(\xi)\,d\xi.}
$$
Applying $L_x$ under the integral gives $f(x)$ by the defining delta identity, while the endpoint values vanish. With the stated continuous coefficients, one may define the normalized distributional equation $G_{xx}+(\beta/\alpha)G_x+(\gamma/\alpha)G=\delta(x-\xi)/\alpha(\xi)$; this avoids multiplying a general distributional second <derivative> by a merely continuous coefficient.
Existence of this inverse <Green function> additionally requires that the homogeneous two-point <boundary value problem> have only the zero solution. The assumptions printed in the question alone do not ensure this: $y''+y$ on $[0,\pi]$ has the nonzero zero-endpoint solution $\sin x$. At such resonance there are <solvability conditions> and no ordinary inverse kernel for arbitrary $f$. The construction below is nonresonant.
Back to article page
