Bernoulli equation
/ ber-NOO-lee /
The Bernoulli equation is the most famous example of a nonlinear equation that is secretly linear in disguise. It looks just out of reach of the integrating-factor method — there is one offending term carrying a power of y — but a single, clever change of variable strips that nonlinearity away and drops the whole thing back into the trustworthy linear toolbox.
Its form is dy/dx + P(x)*y = Q(x)*y^n, where n is any constant other than 0 or 1 (those two values make it already linear). The substitution that saves the day is v = y^(1-n): differentiate and the offending y^n term cancels, converting the equation into a linear first-order equation in v, dv/dx + (1-n)*P(x)*v = (1-n)*Q(x), which you solve by its integrating factor and then undo the substitution to recover y. The recipe is mechanical once you spot the pattern: identify n, set v = y^(1-n), linearize, solve, substitute back. It is the cleanest illustration of the broader strategy of linearizing-by-substitution.
Named for Jacob Bernoulli, who posed it in the 1690s, the equation shows up in real models whose growth term is sublinear or superlinear — certain population and resource models, fluid friction laws, and (with n = 2) the logistic equation can be cast in Bernoulli form. The takeaway worth carrying: a nonlinear equation is not automatically hopeless. Special structures like Bernoulli's admit exact substitutions that recover full, closed-form solutions.
For dy/dx + y = y^2 (so n = 2), set v = y^(1-2) = 1/y. The equation becomes the linear dv/dx - v = -1, with solution v = 1 + C*e^x; back-substituting, y = 1/(1 + C*e^x) — recognizably a logistic-shaped curve.
The substitution v = y^(1-n) turns the Bernoulli equation linear in v.
This Bernoulli is the differential equation of Jacob Bernoulli — not Bernoulli's principle in fluid dynamics, which is a different result by Daniel Bernoulli. Same family name, unrelated statement; do not conflate them.