We consider a differential-boundary equation with algebraic terms on a finite interval \(0 < x < 1\)
\begin{equation}\begin{split}
l(y) \equiv \frac{d}{dx} \biggl( \frac{dy(x)}{dx} &+ \sum_{i=1}^k h_i (x) U_i (y) + \sum_{j=1}^s \lambda_j q_j (x) \biggr)\\
&+ r_1 (x) \frac{dy(x)}{dx} + r_0 (x)y(x) = f(x),
\end{split}\end{equation}
A distinctive feature of these equations is that, alongside the function being sought, a certain number of unknown values must also be determined. This leads to the critical question of unique solvability: how many and what type of conditions need to be imposed on equation (1) to ensure that the resulting problem has a unique solution in a given space? \\
Such equations, consisting of both differential and algebraic parts, are usually called differential-algebraic equations [1,2].
\textbf{Funding}: This research is funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP23485971).
\begin{thebibliography}{6}
\bibitem{1} Kunkel P., Mehrmann V. Differential-Algebraic Equations. Analysis and Numerical Solution, EMS Press, Berlin (2024).
\bibitem{2} Lamour R., Marz R., Weinmuller E.
Boundary-value problems for differential-algebraic equations: a survey,
in Surveys in Differential-Algebraic Equations III,
Springer, Cham (2015), 177--309.
\end{thebibliography}