Toronto Math Forum
APM3462015S => APM346Home Assignments => HA1 => Topic started by: Victor Ivrii on January 20, 2015, 06:47:46 AM

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{eqHA1.2}
\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_x4yu_y=0
\label{eqHA1.3}
\end{equation}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?
c. Explain the difference between (\ref{eqHA1.2}) and (\ref{eqHA1.3}).

not sure this one is right

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.

cï¼‰ 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

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.