# Determine the critical points and identify them as asymptotically stable or unstable? Drawing phase lines?

Here's the question: Determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the $ty$-plane

$$dy/dt = 1 − e^y,\; −∞ < y_0 < ∞.$$

This section of the textbook is all about population growth and is supposed to be in the form $dy/dt=r(1-y/k)y$, so I don't understand how to find the critical pts. in this case. Do I have to actually solve the DE to graph the solutions? Is the phase line just the $y$-axis?

• I edited your question so as to make the $\LaTeX$ work. Cheers! – Robert Lewis Jan 24 '18 at 3:28

The only equilibrium point of $$dy/dt = 1 − e^y,\; −∞ < y_0 < ∞.$$ is at $dy/dt =0.$
Thus equilibrium happens at $$e^y =1$$ that is $y=0$
This equilibrium is asymptotically stable because $y>0 \implies dy/dx <0$ and $y<0 \implies dy/dx>0.$
Therefore the solutions approach the equilibrium solution $y=0$ asymptotically.