PX153 - C4 - separation of variables

if an ODE can be written as $\frac{d y}{d x} = f (x) g (y)$ , then solve by rearranging and integrating:

\int \frac{1}{g (x)} . d y = \int f (x) . d x

eg: solve: $\frac{d N}{d t} = - α N$
$\int \frac{1}{N} . d N = \int - α . d t$
$\ln N = - α t + c$
$N = e^{- α t + c} = e^{c} e^{- α t}$
$∴ N = A e^{- α t}$
- equally valid approach as using a trial approach
eg: solve the differential equation from the height of water example