Applications of Separable ODEs

Orthogonal Trajectories

Take derivative of entire equation
Isolate for $d y / d x$
Translate into new slope ( $y_{n e w}^{'} = - \frac{1}{y_{o l d}^{'}}$ )
Solve for y

Mixing Problems

\frac{d y}{d t} = (rate in) - (rate out)

Generally $y$ represents the amount of stuff in the liquid, with the rate in as a constant, and the rate out in a form of $flow rate \cdot y / initial amount of liquid$

Exponential Growth and Decay

Newton's Law for Cooling

$T_{s} :$ temperature of the surrounding environment
$T (t) :$ temperature at time $t$
Rate of cooling:

\frac{d T}{d t} = k (T - T_{s}), k < 0