$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###[Problems to Chapter 6](id:sect-6.P) > 1. [Problem 1](#problem-6.P.1) > 2. [Problem 2](#problem-6.P.2) > 3. [Problem 3](#problem-6.P.3) > 4. [Problem 4](#problem-6.P.4) > 5. [Problem 5](#problem-6.P.5) **[Problem 1.](id:problem-6.P.1)** a. 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 $u=v/r$) b. 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.](id:problem-6.P.2)** a. 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 $u(r)$? b. 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.](id:problem-6.P.3)** a. 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.](id:problem-6.P.4)** a. 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.](id:problem-6.P.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 $u(x,y)=X(x)Y(y)$. ________________ [$\Uparrow$](../contents.html) [$\uparrow$](./S6.5.html) [$\Rightarrow$](../Chapter7/S7.1.html)