 Solution

Away from $x=\xi$, the Dirichlet Green function solves the homogeneous equation. The endpoint conditions select $A{y}_1(x)$ on the left and $B{y}_2(x)$ on the right. Continuity and the unit derivative jump needed for a Dirac delta function giveWith Wronskian $W=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2\ne 0$, solve to obtain $A=y_2(\xi )/W(\xi )$ and $B=y_1(\xi )/W(\xi )$. ThusThe first derivative term creates no extra delta because $G$ is continuous. The value at $x=\xi$ is the common limiting value.

For the specified constant-coefficient operator, takeTherefore its explicit Green function isFor the forcing step and the homogeneous endpoint conditions inherited from this Green-function problem, $y(x)=2{\int }_{x_0}^{1}G(x,\xi )d\xi$. When $x<x_0$, every integration point lies on the first branch, soWithout those endpoint conditions, an arbitrary homogeneous solution could also be added.