Bernoulli Differential Equations

A rigorous method for Bernoulli equations, including the substitution, excluded cases, and reconstruction checks.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

A Bernoulli equation is $y'+P(x)y=Q(x)y^n$ on an interval, where $n$ is a real constant. When $n\ne0,1$, the nonlinear substitution $v=y^{1-n}$ converts it to the linear equation $v'+(1-n)P(x)v=(1-n)Q(x)$. Cases $n=0$ and $n=1$ are already linear.

Notation and mathematical language

Primes denote differentiation with respect to $x$. From $v=y^{1-n}$, $v'=(1-n)y^{-n}y'$. The transformation is applied on intervals where the powers and division by $y^n$ are defined; zero solutions must be checked separately when the original equation permits them.

Conceptual picture

The substitution works because multiplying the original equation by $(1-n)y^{-n}$ makes the first term exactly $v'$ and the second proportional to $v$. It does not make the original equation linear in $y$; it creates a new dependent variable whose equation is linear.

Fully worked example

Interpretation and application

Bernoulli equations occur when a linear decay or transport term competes with a power-law response. The method gives an exact symbolic solution under its hypotheses. If coefficients come from measured data, that exact formula solves the fitted equation, not the underlying physical system with certainty.