\section*{Problem 6}

\subsection*{a)}
The plot in Figure \ref{fig:eigenvector_10} is showing the discretization of $\hat{x}$ with $n=10$. 
The eigenvectors and corresponding analytical eigenvectors have a complete overlap suggesting the implementation of the algorithm is correct. 
We have included the boundary points for each vector to show a complete solution.
\begin{figure}[h]
    \centering
    \includegraphics[width=0.8\textwidth]{images/eigenvector_10.pdf}
    \caption{The plot is showing the elements of eigenvector $\vec{v}_{1}, \vec{v}_{2}, \vec{v}_{3}$, corresponding to the three lowest eigenvalues of matrix $\boldsymbol{A} (10 \cross 10)$, against the position $\hat{x}$. The analytical eigenvectors $\vec{v}^{(1)}, \vec{v}^{(2)}, \vec{v}^{(3)}$ are also included in the plot.}
    \label{fig:eigenvector_10}
\end{figure}

\subsection*{b)}
For the discretization with $n=100$ the solution is visually close to a continuous curve, with a complete overlap of the analytical eigenvectors, presented in Figure \ref{fig:eigenvector_100}.
\begin{figure}
    \centering
    \includegraphics[width=0.8\textwidth]{images/eigenvector_100.pdf}
    \caption{The plot is showing the elements of eigenvector $\vec{v}_{1}, \vec{v}_{2}, \vec{v}_{3}$, corresponding to the three lowest eigenvalues of matrix $\boldsymbol{A} (100 \cross 100)$, against the position $\hat{x}$. The analytical eigenvectors $\vec{v}^{(1)}, \vec{v}^{(2)}, \vec{v}^{(3)}$ are also included in the plot.}
    \label{fig:eigenvector_100}
\end{figure}