Bernoulli differential equation

In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form y ′ + P ( x ) y = Q ( x ) y n , {\displaystyle y'+P(x)y=Q(x)y^{n},} where n {\displaystyle n} is a real number. Some authors allow any real n {\displaystyle n} , whereas others require that n {\displaystyle n} not be 0 or 1 as they cause the equation to become linear. The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz, who published his result in the same year and whose method is the one still used today. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A notable special case of the Bernoulli equation is the logistic differential equation.