Separation of variables

Occasionally a differential equation allows a separation of variables, which we here exemplify rather than define. The differential equation

$f'(x)=f(x)(1-f(x))$

may be written as

${\frac {dy}{dx}}=y(1-y).$

Pretend that dy and dx are numbers, so that both sides of the equation may be multiplied by dx. Also divide both sides by y(1 − y). We get

${\frac {dy}{y(1-y)}}=dx.$

Integrating both sides, we get

$\int {\frac {dy}{y(1-y)}}=\int dx,$

which, via partial fractions, becomes

$\int {\frac {1}{y}}+{\frac {1}{1-y}}\,dy=\int dx,$

and then

$\log _{e}y-\log _{e}(1-y)=x+C.$

A bit of algebra gives a solution for y:

$y={\frac {1}{1+Be^{-x}}}.$

One may check that if B is any positive constant, this function satisfies the differential equation.