Bernoulli equation
/ ber-NOO-lee /
A Bernoulli equation looks almost like the friendly first-order linear equation, except for one stubborn extra power of y on the right. It is the standard example of a nonlinear equation that is secretly linear in disguise: a single well-chosen substitution strips away the nonlinearity and hands you back a linear equation you can solve by the usual integrating-factor recipe.
Its standard form is y' + p(x) y = q(x) y^n, where n is any real number. The two values n = 0 and n = 1 are already linear, so the interesting cases are everything else. The power y^n is what spoils linearity. The cure is to substitute v = y^(1 - n): a short calculation shows that this new variable obeys a genuinely linear equation, v' + (1 - n) p(x) v = (1 - n) q(x). You solve that linear equation for v(x) by an integrating factor, then undo the substitution to recover y.
Bernoulli equations are named after Jacob Bernoulli, who studied them in the 1690s, and they remain a favourite because they are the cleanest case of 'nonlinear but tameable.' The logistic population model in disguise, and many growth-with-saturation models, are Bernoulli equations — which is part of why the trick is worth knowing rather than just memorizing.
Solve y' + y = x y^2 (here n = 2). Substitute v = y^(1 - 2) = 1/y, so v' = -y'/y^2. The equation becomes v' - v = -x, a linear equation. Its integrating factor is e^(-x); solving gives v = x + 1 + C e^x, so y = 1/(x + 1 + C e^x).
The power y^n vanishes once v = y^(1-n) turns the equation linear.
When n is positive the constant function y = 0 is also a solution, but the substitution v = y^(1-n) divides by y and hides it. Restore y = 0 separately when n > 0.