Bernoulli Equations

A Bernoulli equation is simply a linear first-order equation with one extra term where $y$ is raised to a power. The name just honors Jacob Bernoulli, who posed the problem in 1695; it doesn't describe the equation (and it's a different Bernoulli from the one in fluid flow). These equations show up whenever growth gets slowed by crowding, like a fish population in a lake. All we're really saying is that the equation is almost linear, except for that one annoying power of $y$. The form $y' + p(t)y = q(t)y^n$ looks like a lot, but it's just three pieces: $y'$ (how fast $y$ is changing), $p(t)y$ (the ordinary linear term you saw with integrating factors) and $q(t)y^n$ (the troublemaker, $y$ raised to some power $n$). A change of variable makes the troublemaker go away. It looks complex, but it's just a few simple parts, so bear with it.

Why $v = y^{1-n}$? Divide the whole equation by $y^n$ and you get $y^{-n}y' + p\,y^{1-n} = q$. Now look at the first term. By the chain rule, $(y^{1-n})' = (1 - n)y^{-n}y'$, so $y^{-n}y'$ is just a constant times the derivative of $y^{1-n}$. And the second term already has $y^{1-n}$ in it. So if you call $v = y^{1-n}$, every $y$ disappears and what's left is linear in $v$. (That's also why $n = 0$ and $n = 1$ are excluded: then the equation was linear to begin with.)

Steps:

1. Divide by $y^n$: $y^{-n}y' + p\,y^{1-n} = q$.
2. Let $v = y^{1-n}$, so $v' = (1 - n)y^{-n}y'$, and substitute.
3. Solve the linear equation for $v$ with an integrating factor.
4. Convert back: $y = v^{1/(1-n)}$.

A common mistake. It's tempting to substitute and write $v' + pv = q$, because after dividing by $y^n$ the equation already looks like that. It isn't quite. The first term is $y^{-n}y' = \frac{v'}{1 - n}$, not $v'$, so once you clear that fraction every other term picks up a factor of $(1 - n)$. That's where the $(1 - n)$'s in the formula come from.

You've met one of these already: the logistic equation $P' = rP\left(1 - \frac PK\right) = rP - \frac rK P^2$ is a Bernoulli equation with $n = 2$.

Population models with crowding, some chemical kinetics and certain fluid drag laws are Bernoulli equations; the substitution turns them into textbook linear problems.