Bernoulli differential equation

In mathematics, an ordinary differential equation of the form: In mathematics, an ordinary differential equation of the form: is called a Bernoulli differential equation where n {displaystyle n} is any real number and n ≠ 0 {displaystyle n eq 0} and n ≠ 1 {displaystyle n eq 1} . It is named after Jacob Bernoulli who discussed it in 1695. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A famous special case of the Bernoulli equation is the logistic differential equation. When n = 0 {displaystyle n=0} , the Bernoulli equation is linear. When n = 1 {displaystyle n=1} , it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For n ≠ 0 {displaystyle n eq 0} and n ≠ 1 {displaystyle n eq 1} , the substitution u = y 1 − n {displaystyle u=y^{1-n}} reduces any Bernoulli equation to a linear differential equation. For example, in the case n = 2 {displaystyle n=2} , making the substitution u = y − 1 {displaystyle u=y^{-1}} in the differential equation d y d x + 1 x y = x y 2 {displaystyle {frac {dy}{dx}}+{frac {1}{x}}y=xy^{2}} produces the equation d u d x − 1 x u = − x {displaystyle {frac {du}{dx}}-{frac {1}{x}}u=-x} , which is a linear differential equation. Let x 0 ∈ ( a , b ) {displaystyle x_{0}in (a,b)} and