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.
Referenced By
List of dynamical system and differential equation topics | List of dynamical system topics | List of mathematical topics (S-U)
|
