Key Equations

Key Equations

Euler’s Method

$\begin{array}{c}{x}_n=x_0+nh\hfill \\ y_n=y_{n-1}+hf(x_{n-1},y_{n-1}),\text{where}h\text{is the step size}\hfill \end{array}$

Separable differential equation	$y^{\prime }=f(x)g(y)$
Solution concentration	$\frac{du}{dt}=\text{INFLOW RATE}-\text{OUTFLOW RATE}$
Newton’s law of cooling	$\frac{dT}{dt}=k\left(T-T_s\right)$

Logistic differential equation and initial-value problem	$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right),P\left(0\right)=P_0$
Solution to the logistic differential equation/initial-value problem	$P\left(t\right)=\frac{P_{0}K{e}^{rt}}{\left(K-P_0\right)+P_0{e}^{rt}}$
Threshold population model	$\frac{dP}{dt}=\text{−}rP\left(1-\frac{P}{K}\right)\left(1-\frac{P}{T}\right)$

standard form	$y\prime +p\left(x\right)y=q\left(x\right)$
integrating factor	$\mu \left(x\right)=e^{{\displaystyle \int p\left(x\right)dx}}$