### Review Comments for Number 13

### Dr. Sarah's Comments

Notice that 0 and 3 are stable equilibrium points (solutions tend
toward them from above and below) while 1 is an unstable equilibrium
point.
For the 2nd de, equilibrium points will occur when
the derivative is 0 - is at
x=0, 1-ln(x)=0 or x-3 =0. This gives us equilibrium solutions
at x=0, x=e (2.7....), and x=3. But, notice that ln(0) is undefined,
so there is a problem here with the de. If we zoom in on the dfieldplot
(ie x=2.6..3.1) we find equilibrium solutions at x=e and x=3.
Zooming in again, we see that x=3 is stable, while x=e is unstable.

In part c, Change the plot command to:
plot(sol1,x=.01..4);

In part d, notice that limit of the rhs of the de (as x approaches
0 from the right) is 0, so we could consider x=0 an equilibrium
point. Notice that the rhs of the de is negative for x >0,
as x approaches 0. Hence, solutions starting about x=0 decrease to
x=0, and so it is a stable equilibrium solution.

What about x=e? The rhs of the d.e. is negative for solutions
below x=e, and positive for solutions just above x=e.
Hence, those below x=e decrease away from x=e, and those just above
x=e increase away from x=e. Hence this is an unstable equilibrium
solution

What about x=3? The rhs of the d.e. is positive for solutions
just below x=3, and negative for solutions just above x=3.
Hence, those just below x=3 increase towards x=3, while those
above x=3 decrease towards x=3. Hence this is a stable
equilibrium solution.

This analysis matches our graphical evidence.

notice that dfieldplot has prolems in the interval between e and 3
because the right hand side of the derivative is so small (positive),
as we can see in the plot of this rhs (it bumps to a positive curve,
but stays very close to 0). If we don't zoom in on the dfieldplot
enough, then the smallness of the rhs of the de makes it appear that there
are more equilibrium solutions, when comparing to the dfieldplot
away from this problem area. Zooming in on just this area corrects
for this.

### Peer Review Comments

not here? What was initial problem - need more comments in Maple work.