We discuss existence and regularity of bounded variation solutions of the Dirichlet problem for the one-dimensional capillarity-type equation
\begin{equation*}
\Big( u'/{ \sqrt{1+{u'}^2}}\Big)'
= f(t,u) \quad \hbox{ in } {]-r,r[},
\qquad
u(-r)=a, \, u(r) = b.
\end{equation*}
We prove interior regularity of solutions and we obtain a precise description of their boundary behaviour. This is achieved by a direct and elementary approach that exploits the properties of the zero set of the right-hand side $f$ of the equation.
