In mathematics, an ordinary differential equation of the form
is called a Bernoulli equation when n≠1, 0, which is named after Jakob Bernoulli, who discussed it in 1695 (Bernoulli 1695). Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.
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Dividing by yields
A change of variables is made to transform into a linear first-order differential equation.
The substituted equation can be solved using the integrating factor
Consider the Bernoulli equation
We first notice that is a solution. Division by yields
Changing variables gives the equations
which can be solved using the integrating factor
Multiplying by ,
Note that left side is the derivative of . Integrating both sides results in the equations
The solution for is
as well as .
Verifying using MATLAB symbolic toolbox by running
x = dsolve('Dy-2*y/x=-x^2*y^2','x')
gives both solutions:
0 x^2/(x^5/5 + C1)
also see a solution by WolframAlpha, where the trivial solution is missing.