 Solution

For a fixed source point $\xi \in (a,b)$, the Green function is defined bywhere $\delta$ is the Dirac delta distribution and the operator acts on $x$. The corresponding solution isApplying $L_x$ 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: $y^{\prime \prime }+y$ 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.