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###[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)