We prove existence, uniqueness and stability of solutions of the prescribed curvature problem
\begin{equation*}
\begin{cases}
\bigl({u'}/{\sqrt{1 + u'^2}}\bigr)' = au -{b}/{\sqrt{1 + u'^2}} \quad \text{in }[0,1]\\
u'(0)=u(1)=0,
\end{cases}
\end{equation*}
for any given $a>0$ and $b>0$.
We also develop a linear monotone iterative scheme for approximating the solution. This equation has been proposed as a model of the corneal shape in the recent paper \cite{OkPl}, where a simplified version obtained by partial linearization has been investigated.
