$\newcommand{\const}{\mathrm{const}}$

Deadline Monday, September 24, 9 pm

- Find general solution \begin{equation} u_x-3u_y=0;\label{eq-1} \end{equation}
- Solve IVP problem $u|_{x=0}=e^{-y^2}$ for equation (\ref{eq-1}) in $\mathbb{R}^2$;
- Consider equation (\ref{eq-1}) in $\{x>0, y>0\}$ with the initial condition $u|_{x=0}=y$ ($y>0$); where this solution defined? Is it defined everywhere in $\{x>0, y>0\}$ or do we need to impose condition at $y=0$? In the latter case impose condition $u|_{y=0}=x$ ($x>0$) and solve this IVBP;
- Consider equation (\ref{eq-1}) in $\{x<0, y>0\}$ with the initial condition $u|_{x=0}=y$ ($y>0$); where this solution defined? Is it defined everywhere in $\{x<0, y>0\}$ or do we need to impose condition at $y=0$? In the latter case impose condition $u|_{y=0}=x$ ($x<0$) and solve this IVBP.

- Find the general solution of \begin{equation} xu_x+4yu_y=0\label{eq-2} \end{equation} in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?
- Find the general solution of \begin{equation} xu_x-4yu_y=0\label{eq-3} \end{equation} in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?
- Explain the difference.

Find the solution of \begin{equation} \left\{\begin{aligned} &u_x+3u_y=xy,\\ &u|_{x=0}=0. \end{aligned} \right.\label{eq-4} \end{equation}

- Find the general solution of \begin{equation} yu_x-xu_y=xy;\label{eq-5} \end{equation}
- Find the general solution of \begin{equation} yu_x-xu_y=x^2+y^2;\label{eq-6} \end{equation}
- In one instanse solution does not exist. Explain why.

- Find the general solution of \begin{equation} u_{tt}-9u_{xx}=0;\label{eq-7} \end{equation}
- Solve IVP \begin{equation} u|_{t=0}=x^2,\quad u_t|_{t=0}=x \end{equation} for (\ref{eq-7});
- Consider (\ref{eq-7}) in $\{x>3t, x>-3t\}$ and find a solution to it, satisfying Goursat problemCorrected September 23 \begin{equation} u|_{x=3t}=t,\quad u|_{x=-3t}=2t. \end{equation}

**Derivation of a PDE describing traffic flow.**
The purpose of this problem is to derive a model PDE that describes a congested one-dimensional highway. Let

- $\rho(x,t)$ denote the traffic density : the number of cars per kilometer at time $t$ located at position $x$;
- $q(x,t)$ denote the traffic flow: the number of cars per hour passing a fixed place $x$ at time $t$;
- $N(t,a,b)$ denote the number of cars between position $x=a $ and $ x=b$ at time $t$.
- Derive a formual for $N(t,a,b)$ as an integral of the traffic density. You can assume there are no exits or entrances between position $a$ and $b$.
- Derive a formula for $\frac {\partial N} {\partial t}$ depending on the traffic flow. '''Hint:''' You can express the change in cars between time $t_1=t$ and $t_2= t+h$ in terms of of traffic flow;
- Differentiate with respect to $t$ the integral form for $N$ from part (a) and make it equal to the formula you got in part (b). This of the integral form of conservation of cars;
- Express the right hand side of the formula of part (c) in terms of an integral. Since $a,b$ are arbitrary, obtain a PDE. This PDE is called the conservation of cars equation;
- What equation do you get in part (4) if $ q=c \rho$, for some constant $c$. What choice of $c$ would be more realistic, i.e. what should $c$ be function of?