$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
Solve equation graphically
means that you plot a corresponding
function and points $(z_n,0)$ where it intersects with $OX$ will
give us all the frequencies $\omega_n=\omega (z_n)$.
Simple solution
$u(x,t)=X(x)T(t)$.
You may assume that all eigenvalues are real (which is the case).
Problem 1. Justify Example 4.2.6 and Example 4.2.7: Consider eignevalue problem with Robin boundary conditions \begin{align} & X'' +\lambda X=0 && 0< x< l,\label{p}\\[3pt] & X'(0)=\alpha X(0),\label{q}\\ & X'(l)=-\beta X(l),\label{r} \end{align} with $\alpha, \beta \in \mathbb{R}$.
Problem 2. Analyse the same problem albeit with Dirichlet condition on the left end: $X(0)=0$.
Problem 3. Oscillations of the beam are described by equation \begin{equation} u_{tt} + K u_{xxxx}=0, \qquad 0 < x < l. \label{a} \end{equation} with $K >0$.
If both ends clamped (that means having the fixed positions and directions) then the boundary conditions are \begin{align} &u(0,t)=u_{x}(0,t)=0,\label{b}\\ &u(l,t)=u_{x}(l,t)=0.\label{c} \end{align}
Hint. One can change coordinate system so that interval becomes $[-L,L]$ with $L=l/2$ and then consider separately even and odd eigenfunctions.
Problem 4. Consider oscillations of the beam with both ends simply supported: \begin{align} &u(0,t)=u_{xx}(0,t)=0,\label{d}\\ &u(l,t)=u_{xx}(l,t)=0.\label{e} \end{align} Follow the previous problem (all parts) but also consider eigenvalue $0$.
Problem 5. Consider oscillations of the beam with both ends free: \begin{align} &u_{xx}(0,t)=u_{xxx}(0,t)=0,\label{f}\\ &u_{xx}(l,t)=u_{xxx}(l,t)=0.\label{g} \end{align} Follow the previous problem but also consider eigenvalue $0$.
Problem 6. Consider oscillations of the beam with the clamped left end and the simply supported right end. Then boundary conditions are (\ref{a}) and (\ref{e}).
Note. In this case due to the lack of symmetry you cannot consider separately even and odd eigenfunctions.
Problem 7. Consider oscillations of the beam with the clamped left end and the free right end. Then boundary conditions are (\ref{a}) and (\ref{g}).
Problem 8. Consider oscillations of the beam with the simply supported left end and the free right end. Then boundary conditions are (\ref{d}) and (\ref{g}).
Problem 9.
Consider wave equation with the Neumann boundary condition on the left
and weird
b.c. on the right:
\begin{align}
& u_{tt}-c^2u_{xx}=0 && 0< x< l, \\
& u_x (0,t)=0, \\
& (u_x + i \alpha u_t) (l,t)=0
\end{align}
with $\alpha \in \mathbb{R}$.
weirdeigenvalue problem for ODE;
Hint. You may assume that all eigenvalues are real (which is the case).
Problem 10.
Consider energy levels of the particle in the rectangular well
\begin{equation}
-u_{xx}+V u =\lambda u
\end{equation}
with $V(x)=\left\{\begin{aligned} -&H && |x|\le L,\\
&0 &&|x|>0\end{aligned}\right.$
Hint. Solve equation for $|x|< L$ and for $|x| > L$ and solution must be continous (with its first derivative) as $|x|=L$: $u(L-0)=u(L+0)$, $u_x (L-0)=u_x (L+0)$ and the same at $-L$.
Hint. All eigenvalues belong to interval $(-H,0)$.
Hint. Consider separately even and odd eigenfunctions.