## Bernoulli Differential Equations

Bernoulli differential equations have the form:

where n represents a real number. For n = 0, Bernoulli's equation reduces to a linear first-order differential equation.

Bernoulli differential equations may be solved by initially mulitplying both sides by y

Substituting w = y

Bernoulli differential equations may be solved by initially mulitplying both sides by y

^{-n}:Substituting w = y

^{1-n}(with w' = (1 - n)y^{-n}y' ), the above equation becomes:The above equation may be solved for w(x) using techniques for linear differential equations and solving for y.

Example:

Solve the equation y' + xy = xy

^{3}Solution: