= Bernoulli differential equation
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{title2=$y'+P(x)y=Q(x)y^n$}
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A <Bernoulli differential equation> is a first-order <ordinary differential equation> of the displayed form, with $P$ and $Q$ <continuous functions> on an <interval> and a fixed real exponent $n$. A nonlinear power can hide a <first-order linear differential equation>: on a suitable branch, the <Bernoulli power substitution> $z=y^{1-n}$ removes it exactly. The transformed <solution of a differential equation> can be completely regular even when recovering $y$ produces <finite-time blowup>.
For example, the <reciprocal Bernoulli equation with finite-time blowup> $y'+y=e^xy^2$ becomes $z'-z=-e^x$ under $z=1/y$. The <initial conditions> $y(0)=1/2$ and $y(0)=-1$ select a positive trajectory blowing up at $x=2$ and a negative trajectory tending to zero as $x\to\infty$, respectively. The <integrating factor> exposes both behaviors with the same short calculation.
The cases $n=0$ and $n=1$ are already <first-order linear differential equations>: respectively $y'+Py=Q$, with the zeroth-power term interpreted as the constant $1$, and $y'+(P-Q)y=0$. There is no need for a nonlinear change of variable; for $n=1$, $y^{1-n}=1$ would discard the unknown entirely. For other exponents, the substitution must respect the domain of <exponentiation> and may lose the zero <solution of a differential equation>.<Notes on Diffy Qs: Differential Equations for Engineers by Jiří Lebl/1.5 Substitution>[Notes on Diffy Qs, §1.5.2 Bernoulli equations: substitution v equals y to the power 1 minus n; n equals 0 or 1 is already linear; Example 1.5.1 selects the positive root using its initial condition]{ref}
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