# Toronto Math Forum

## APM346-2015S => APM346--Home Assignments => HA1 => Topic started by: Victor Ivrii on January 20, 2015, 06:47:46 AM

Title: HA1 problem 2
Post 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{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}).
Title: Re: HA1 problem 2
Post by: Biao Zhang on January 22, 2015, 10:18:30 PM
not sure this one is right
Title: Re: HA1 problem 2
Post by: Jessica Chen on January 22, 2015, 10:51:12 PM
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.
Title: Re: HA1 problem 2
Post by: Ping Wei on January 23, 2015, 10:20:06 AM
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
Title: Re: HA1 problem 2
Post by: Victor Ivrii on January 23, 2015, 11:02:17 AM
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.