Approximate solution of a boundary value problem with a discontinuous solution

Using spline-functions for three-point rational interpolants an approximate solution of the boundary value problem: $y^\prime +p(x) y=f(x)$, $y(a)=A$, $y(b)=B$ is constructed. In this case, the functions $p(x)$ and $f(x)$ are assumed to be continuous on the segment $[a,b]$ and it is allowed, that there exists a solution $y (x)$ that can have a discontinuity of the first kind with a jump at a given point $\tau\in (a, b)$.