Bernoulli Differential Equation: What It Is and How to Solve It

Jacob Bernoulli, not Daniel’s principle

The Bernoulli differential equation appears in courses on ordinary differential equations and carries the name of the Swiss mathematician Jacob Bernoulli (1654-1705). It is a topic in mathematics: the Bernoulli principle in fluid physics belongs to Daniel Bernoulli, a relative of Jacob, and shares only the surname with this equation. For the broader picture of this type of equation, before the specific case covered in this Recap, see differential equations.

The general form and the role of the exponent n

According to Paul’s Online Math Notes (Lamar University), Wolfram MathWorld, and Mathematics LibreTexts, the Bernoulli equation is written y’ + p(x)y = q(x)y^n, with p(x) and q(x) continuous functions on the interval considered and n any real number: whole, fractional, or negative. The exponent n determines whether the equation is already linear. When n equals 0 or 1, the equation is a special case of the first-order linear equation (and for n = 1 it is also separable), solvable with the standard methods of calculus: here the substitution is not needed. For any other value of n the equation is nonlinear, which is where the substitution technique described below comes in.

The method, step by step

The procedure reduces the nonlinear equation to a linear equation in a different function. It starts by dividing the whole equation by y^n: the result is y^(-n)y’ + p(x)y^(1-n) = q(x), a step of pure symbolic manipulation similar to what is covered in algebra. At this point the substitution v = y^(1-n) is introduced. Differentiating v with respect to x gives v’ = (1-n)y^(-n)y’, a relation that lets the derivative term be replaced. The equation becomes a first-order linear equation in the function v, in the form v’ + (1-n)p(x)v = (1-n)q(x): the notion of linearity here is the same one studied in linear algebra, every term first-degree in the unknown and its derivative. From here it is solved like any linear equation, with the integrating factor. Once the linear equation is solved, substituting back v = y^(1-n) gives y, the function sought from the start.

An example with a closed-form solution

Worked example: Paul’s Online Math Notes solves y’ + (4/x)y = x³y², with the initial condition y(2) = -1 and x > 0. Here n = 2. Dividing by y² gives y^(-2)y’ + (4/x)y^(-1) = x³; with v = y^(-1) and v’ = -y^(-2)y’, the equation becomes v’ - (4/x)v = -x³, linear in v. The integrating factor is x^(-4), and the solution in v comes out as v(x) = cx⁴ - x⁴ ln x. Returning to y (v = y^(-1)) gives y^(-1) = x⁴(c - ln x). The initial condition y(2) = -1 fixes c = ln 2 - 1/16, and the final solution is y(x) = -16 / [x⁴(1 + 16 ln(x/2))].

The example shows the full sequence: divide, substitute, solve the linear equation with the integrating factor, return to y, and use the initial condition only at the end to fix the constant.

An example without a closed-form solution

Worked example: Mathematics LibreTexts solves xy’ + y(x+1) + xy⁵ = 0, with y(1) = 1. Here n = 5, and the substitution is v = y^(-4), with v’ = -4y^(-5)y’. After the substitution and rearranging terms, the equation becomes v’ - [4(x+1)/x]v = 4, still linear in v. The source notes that the integral needed to solve this linear equation has no elementary closed form, so the solution stays written with that integral.

The method works the same way every time, even when the final result stays awkward. The substitution always reduces the original equation to a linear one; whether that linear equation, once solved, turns back into a simple formula in y depends on the original equation itself.

Where the name comes from

Wolfram MathWorld places the Bernoulli equation among the standard reference texts on differential equations: a textbook topic, not an isolated result. Mathematics LibreTexts notes that the name “Bernoulli equation” has been used in the literature for more than one equation, part of why it gets confused with the fluid physics principle. That physics principle, due to Daniel Bernoulli, describes the behavior of a moving fluid and is not the differential equation covered here, which instead carries the name of Jacob Bernoulli, another member of the same family of Swiss mathematicians and physicists.