$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
Problem 1.
Find the solutions that depend only on $r$ of the equation \begin{equation*} \Delta u:=u_{xx}+u_{yy}+u_{zz}=k^2u, \end{equation*} where $k$ is a positive constant. (Hint: Substitute $u=v/r$)
Find the solutions that depend only on $r$ of the equation \begin{equation*} \Delta u:=u_{xx}+u_{yy}+u_{zz}=-k^2u, \end{equation*} where $k$ is a positive constant. (Hint: Substitute $u=v/r$)
Problem 2.
Try to find the solutions that depend only on $r$ of the equation \begin{equation*} \Delta u:=u_{xx}+u_{yy}=k^2u, \end{equation*} where $k$ is a positive constant. What ODE should satisfy $u(r)$?
Try to find the solutions that depend only on $r$ of the equation \begin{equation*} \Delta u:=u_{xx}+u_{yy}=-k^2u, \end{equation*} where $k$ is a positive constant. What ODE should satisftfy $u(r)$?
Problem 3.
Solve \begin{align*} & \Delta u:=u_{xx}+u_{yy}=0&& \text{in } r<a\\[3pt] & u|_{r=a}=f(\theta). \end{align*} where we use polar coordinates $(r,\theta)$ and $f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.$
Solve \begin{align*} & \Delta u:=u_{xx}+u_{yy}=0&& \text{in } r>a\\[3pt] & u|_{r=a}=f(\theta),\\[3pt] & \max |u| <\infty. \end{align*} where we use polar coordinates $(r,\theta)$ and $f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.$
Problem 4.
Solve \begin{align*} & \Delta u:=u_{xx}+u_{yy}=0&& \text{in } r<a\\[3pt] & u_r|_{r=a}=f(\theta) \end{align*} where we use polar coordinates $(r,\theta)$ and $f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.$
Solve \begin{align*} & \Delta u :=u_{xx}+u_{yy}=0&& \text{in } r>a\\[3pt] & u_r|_{r=a}=f(\theta),\\[3pt] & \max |u| <\infty. \end{align*} where we use polar coordinates $(r,\theta)$ and $f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.$
Problem 5. Describe all real-valued solutions of biharmonic equation \begin{equation} u_{xxxx}+2u_{xxyy}+u_{yyyy}=0 \end{equation} which one can obtain by a method of separation $u(x,y)=X(x)Y(y)$.