On solvability of certain classes of irregular the second order variation problems

In the work the irregular model $-p(x)u'(x)+p(+0)u'(+0)+\int_0^xu(\tau)\,dQ(\tau)=F(x)-F(0)$ of the Stiltjes string on segment $[0,\ell]$ with boundary conditions $u(0)=u(\ell)=0$ is discussed. The solvability conditions of the mentioned problem are described.