Bernoulli Differential Equations

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.

Conditions and key results

$P$ and $Q$ should be continuous on the solution interval. For non-integer powers, real-valued solutions may require a sign or positivity restriction. Reconstructing $y=v^{1/(1-n)}$ can introduce branches, so the initial condition and real domain determine the valid branch.

A reliable strategy

1. Put the equation in $y'+P(x)y=Q(x)y^n$ form and identify $n$.
2. Record any zero solution and domain restrictions before multiplying by $y^{-n}$.
3. Set $v=y^{1-n}$, derive the linear $v$ equation, and solve it with an integrating factor.
4. Transform back to $y$, apply the initial condition, and substitute into the original nonlinear equation.

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.

Common mistakes

Verification and reasonableness

• Differentiate the reconstructed $y$ and use the original, not transformed, equation for verification.
• Check that $v=y^{1-n}$ holds on the chosen branch.
• Locate denominator zeros or branch endpoints and report the maximal interval through the initial point.

Practice

1. What substitution linearizes $y'+P y=Qy^3$?
2. What equation results from $y'+Py=Qy^n$?
3. Why check $y=0$ separately?

Further deduction

A useful special form is the logistic equation $y'=ry-(r/K)y^2$, which is Bernoulli with $n=2$ after writing $y'-ry=-(r/K)y^2$. The substitution $v=1/y$ yields $v'+rv=r/K$, a linear equation. This connects the Bernoulli technique with population models while also showing why the equilibrium $y=0$ disappears during division and must be restored separately.

Related topics

• First-Order Linear Differential Equations
• Separable Differential Equations
• Existence and Uniqueness