Key Equations

Euler’s Method

\begin{matrix} x_{n} = x_{0} + n h \\ y_{n} = y_{n - 1} + h f (x_{n - 1}, y_{n - 1}), where h is the step size \end{matrix}

Separable differential equation	$y^{'} = f (x) g (y)$
Solution concentration	$\frac{d u}{d t} = INFLOW RATE - OUTFLOW RATE$
Newton’s law of cooling	$\frac{d T}{d t} = k (T - T_{s})$

Logistic differential equation and initial-value problem	$\frac{d P}{d t} = r P (1 - \frac{P}{K}), P (0) = P_{0}$
Solution to the logistic differential equation/initial-value problem	$P (t) = \frac{P_{0} K e^{r t}}{(K - P_{0}) + P_{0} e^{r t}}$
Threshold population model	$\frac{d P}{d t} = − r P (1 - \frac{P}{K}) (1 - \frac{P}{T})$

standard form	$y' + p (x) y = q (x)$
integrating factor	$μ (x) = e^{\int p (x) d x}$