Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation.
The most important such equation has the form y' + p(t)y = q(t) yⁿ and is called Bernoulli's equation after Jakob Bernoulli. If n ≠ 0, 1, then the substitution v = y¹⁻ⁿ reduces Bernoulli's equation to a linear equation.
Solve the given Bernoulli equation by using this substitution:
y' = εy − σy⁵, ε > 0 and σ > 0.
This equation occurs in the study of the stability of fluid flow.