Separation of variables

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

may be written as

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

At this point we have separated the variables x and y from each other, since x appears only on the right side of the equation and y only on the left.

Integrating both sides, we get

which, via partial fractions, becomes

and then

A bit of algebra gives a solution for y:

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

This process also exemplifies the utility of the Leibniz notation, in which dy and dx are thought of as infinitely small increments of y and x respectively.