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.