On the solvability of a singular boundary-value problem for the equation $f(t,x,x',x'')=0$

In this work we consider boundary value problems of the form
\begin{gather*} f(t,x,x',x'')=0,\quad 0<t<1,\\ x(0)=0,\quad x'(1)=b,\quad b>0, \end{gather*}
where the the scalar function $f(t,x,p,q)$ may be singular at $x=0$. As far as we know, the solvability of the singular boundary value problems of this form has not been treated yet. Here we try to fill in this gap. Examples, illustrating our main result, are included.