HA1 problem 2

[Victor Ivrii]

Solutions to be posted as a "Reply" only after January 22, 21:00

a. Find the general solution of
\begin{equation}
xu_x+4 yu_y=0
\label{eq-HA1.2}
\end{equation}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?

b. Find the general solution of
\begin{equation}
xu_x-4yu_y=0
\label{eq-HA1.3}
\end{equation}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?

c. Explain the difference between (\ref{eq-HA1.2}) and (\ref{eq-HA1.3}).

[Biao Zhang]

not sure this one is right

[Jessica Chen]

Not sure if I interpreted it right either.

c.
The difference between a and b is
part a) the characteristic lines look like a parabola, all trajectories have (0, 0) as the limit point;
part b) the characteristic lines look like a delta function, only (x=0, y=0) has (0,0) as the limit point.

[Ping Wei]

c） The difference between two cases is that in one of them all trajectories have (0,0) as the limit points and in another only those with x=0 or y=0

[Victor Ivrii]

Good!

Biao, math operators like sin, cos, ln etc should be typed upright and have a horizontal thin space after; in LaTeX it is achieved by putting backslash in front: \ln x. Observe, the space after (to avoid "undefined" error)

I just concretize c: it is node vs saddle (remember ODE class?). So solution in (a) is $f(y/x^4)$ and it is continuous iff it is constant; solution in (b) is $f(yx^4)$ and it is  continuous iff $f$ is continuous.