I cannot remember ever having to solve a Bernoulli differential equation outside of a formal course in ordinary differential equations, but I found this solution on an old thumb drive, and I happen to have written it in html format, so now you can enjoy it, too.
A nonlinear ODE of the form
Can be reduced to a linear differential equation using the substitution
Where n is any real number. Of course if n=1 or n=0, then the equation is already linear and no substitution is necessary.
Solving for dy/du results in
By the chain rule the first derivative of y with respect to x can be written as
And so using (3) the first derivative of y can be rewritten as
Substituting (4) into Bernoulli’s equation produces the rather spongy looking expression
But when you multiply both sides by something interesting emerges.
Which is the same as the original substitution that
So (5) is actually just a regular old first order linear differential equation that can be solved using any known technique. In fact (5) is just the familiar standard form of a linear equation. To see this let
then (5) becomes
Divide both sides by x to get the form shown in (1).
Substituting P(x) and Q(x) into (6) gives
The integrating factor for this last equation on, say, (0,∞) is
Multiplying both sides of (7) by this result gives
After integrating both sides you have
Finally back substituting yields a solution to y