The Bernoulli differential equation is y^t + p(x)y = q(x)y^n. On the interval being worked on, p(x) and q(x) are continuous functions, and n is a real number.
If n = 0 or n = 1, then the equation is linear. For n values other than zero or one, the differential equation should be divided by y^n to yield the equation y^-n y^t + p(x)y^1-n = q(x). To yield an equation that can be solved, v will replace y^1-n to convert the differential equation in terms of v. Taking the derivative of this leads to v^t = (1 - n)y^n y^t. Plugging this in along with the substitution gives the equation 1/1-n(v^t) + p(x)v = q(x). It is now a linear differential equation that can be solved for v, which can be plugged into the original equation to solve for y. This solves the original Bernoulli differential equation.