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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?

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

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