Bring the chapter together: separate the variables and integrate, find the equilibria of an autonomous equation and read its phase line, test an equilibrium for stability with one derivative, and use the exponential and logistic models. No worked example sits above the answers.

A navigator carries a chart, a compass and a clock, and the skill is knowing which of them the question in front of the ship needs.

Decide first what is asked. A separable equation $y'=g(x)h(y)$ is solved by putting each variable on its own side and integrating, after setting aside the constant solutions where $h=0$. An autonomous equation $y'=f(y)$ is read without solving it: its equilibria are the zeros of $f$, and the sign of $f$ between them is the direction of the phase line.

• Dividing by $h(y)$ loses the solutions where $h(y)=0$: check them separately.
• $f(c)=0$ and $f'(c)<0$: stable. $f'(c)>0$: unstable. $f'(c)=0$: read the phase line.
• $y'=ky$ gives $y=y_{0}e^{kt}$: growth when , decay when .
• The logistic equation $y'=ry\bigl(1-\tfrac{y}{K}\bigr)$ has an unstable equilibrium at and a stable one at the carrying capacity .