$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

"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 $\alpha, \beta \in \mathbb{R}$.

**Prove**that positive eigenvalues are $\lambda_n=\omega_n^2$ and the corresponding eigenfunctions are $X_n$ where $\omega_n>0$ are roots of \begin{align} & \tan (\omega l)= \frac{(\alpha+\beta)\omega}{\omega^2-\alpha\beta};\\ & X_n= \omega_n \cos (\omega_n x) +\alpha \sin (\omega_n x); \end{align} with $n=1,2,\ldots$.**Solve**this equation graphically.Prove that negative eigenvalues if there are any are $\lambda_n=-\gamma_n^2$ and the corresponding eigenfunctions are $Y_n$ where $\gamma_n>0$ are roots of \begin{align} & \tanh (\gamma l )= {-\frac{(\alpha + \beta)\gamma } {\gamma ^2 + \alpha\beta}},\\ & Y_n(x) = \gamma_n \cosh (\gamma_n x) + \alpha \sinh (\gamma_n x). \end{align}

**Solve**this equation graphically.To investigate how many negative eigenvalues are, consider the threshold case of eigenvalue $\lambda=0$: then $X=cx+d$ and plugging into b.c. we have $c=\alpha d$ and $c=-\beta (d+lc)$; this system has non-trivial solution $(c,d)\ne 0$ iff $\alpha+\beta+\alpha \beta l =0$. This hyperbola divides $(\alpha,\beta)$-plane into three zones.

**Prove**that eigenfunctions corresponding to different eigenvalues are orthogonal: \begin{equation} \int_0^l X_n(x)X_m (x)\,dx =0\qquad\text{as } \lambda_n\ne \lambda_m \end{equation} where we consider now all eigenfunctions (no matter corresponding to positive or negative eigenvalues).**Bonus**Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional.

**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}

**Find**equation describing frequencies and corresponding eigenfunctions (You may assume that all eigenvalues are real and positive).**Solve**this equation graphically.**Prove**that eigenfunctions corresponding to different eigenvalues are orthogonal.**Bonus**Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional.

*Hint.* Change coordinate system so that interval becomes $[-L,L]$ with $L=l/2$; consider separately even and odd eigenfunctions.

**Problem 4.**
Consider oscillations of the beam with both ends free:
\begin{align}
&u_{xx}(0,t)=u_{xxx}(0,t)=0,\label{d}\\
&u_{xx}(l,t)=u_{xxx}(l,t)=0.\label{e}
\end{align}
Follow previous problem but also consider eigenvalue $0$.

**Problem 5.**
Consider oscillations of the beam with the clamped left end and the free 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 6.**
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 $\alpha \in \mathbb{R}$.

**Separate**variables;**Find**"weird" eigenvalue problem for ODE;**Solve**this problem;**Find**simple solution $u(x,t)=X(x)T(t)$.

*Hint.* You may assume that all eigenvalues are real (which is the case).

**Problem 7.**
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.