Revision sheet

Euler's method

$y_{n+1} = y_n + h\,f(x_n, y_n)$: slope at the current point × step.

Watch out: update the slope at each new point.

Logistic growth

• $P' = rP(1 - P/K)$; equilibria $0$ and $K$.
• $P \to K$; fastest growth at $K/2$.

Watch out: $r$ is a rate, $K$ is a population.

The logistic solution

$P(t) = \dfrac{K}{1 + Ae^{-rt}}$, $A = \dfrac{K - P_0}{P_0}$, $P \to K$.

Watch out: check $A$ by computing $P(0)$.