Draw characteristics and find the general solution to each of the following equations
\begin{gather}
2 u_t +3u_x=0;\\
u_t + t u_x=0;\\
u_t - t u_x=0;\\
u_t+ t^2 u_x=0;\\
u_t +x u_x=0;\\
u_t +t x u_x=0;\\
u_t+x^2u_x=0;\\
u_t+(x^2+1)u_x=0;\\
u_t+(t^2+1)u_x=0;\\
(x+1)u_t +u_x=0;\\
(x+1)^2u_t+u_x=0;\\
(x^2+1)u_t+u_x=0;\\
(x^2-1)u_t+u_x=0.
\end{gather}
Consider IVP problem $u|_{t=0}=f (x)$ as $-\infty < x < \infty$; does solution always exists? If not, what conditions should satisfy $f(x)$? Consider separately $t > 0$ and $t < 0$.
Where this solution is uniquely determined? Consider separately $t > 0$ and $t < 0$.
Consider this equation in $\{t > 0, x > 0\}$ with the initial condition $u|_{t=0}=f(x)$ as $x > 0$; where this solution defined? Is it defined everywhere in $\{t > 0, x > 0\}$ or do we need to impose condition at $x=0$? In the latter case impose condition $u|_{x=0}=g(t)$ ($t > 0$) and solve this IVBP;
Consider this equation in $\{t > 0, x < 0\}$ with the initial condition $u|_{t=0}=f(x)$ as $x < 0$; where this solution defined? Is it defined everywhere in $\{t > 0, x < 0\}$ or do we need to impose condition at $x=0$? In the latter case impose condition $u|_{x=0}=g(t)$ ($t > 0$) and solve this IVBP;
Consider problems (d) as $t < 0$;
Consider problems (e) as $t < 0$;
Problem 2.
Find the general solution to each of the following equations
\begin{gather}
xu_x+ yu_y=0,\\
xu_x-yu_y=0
\end{gather}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$? Explain the difference between these two cases;
Find the general solution to each of the following equations
\begin{gather}
yu_x+ xu_y=0,\\
yu_x-xu_y=0
\end{gather}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?
Explain the difference between these two cases;
Problem 3.
In the same way consider equations
\begin{gather}
(x^2+1)yu_x+(y^2+1)xu_y=0;\\
(x^2+1)yu_x-(y^2+1)xu_y=0.
\end{gather}
Problem 4.
Find the solutions of
\begin{align}
&\left\{\begin{aligned}
&u_x+3u_y=xy,\\
&u|_{x=0}=0;
\end{aligned} \right.\\[3pt]
&\left\{\begin{aligned}
&u_x+3u_y=u,\\
&u|_{x=0}=y.
\end{aligned} \right.
\end{align}
Problem 5.
Find the general solutions to each of
\begin{gather}
yu_x-xu_y=x;\\[3pt]
yu_x-xu_y=x^2;\\[3pt]
yu_x+xu_y=x;\\[3pt]
yu_x+xu_y=x^2.
\end{gather}
In one instance solution does not exist. Explain why.
Problem 6.
Solve IVP
\begin{equation}
\left\{\begin{aligned}
&u_t+uu_x=0,\qquad t > 0;
\\
&u|_{t=0}=f(x)
\end{aligned}\right.
\end{equation}
and describe domain in $(x,t)$ where this solution is properly defined
with one of the following initial data
\begin{align}
f(x)&=\hphantom{-}\tanh (x);
\\[5pt]
f(x)&=-\tanh (x);\\
f(x)&=\left\{\begin{aligned}
-1& && x < -a,\\
x/a& && -a\le x \le a,\\
1& && x > a;
\end{aligned}\right.
\\
f(x)&=\left\{\begin{aligned}
1& && x < -a,\\
-x/a& && -a\le x \le a,\\
-1& && x > a;
\end{aligned}\right.
\\
f(x)&=\left\{\begin{aligned}
-1& && x<0,\\\\
1& && x> 0;\\
\end{aligned}\right.
\\
f(x)&=\hphantom{-}\sin(x).
\\
f(x)&=\left\{\begin{aligned}
&\sin (x) && |x| < \pi,\\
&0 && |x| > \pi,
\end{aligned}\right.\\
f(x)&=\left\{\begin{aligned}
&-\sin (x) && |x| < \pi,\\
&0 && |x| > \pi,
\end{aligned}\right.
\end{align}
Here $a > 0$ is a parameter.