Bernoulli Equation Solver

y' + Py = Qyⁿ solver

CalculatorsFreeNo Signup
4.5(142 reviews)
All Tools

Loading tool...

About Bernoulli Equation Solver

A Bernoulli equation solver for ODEs of the form y' + P(x)y = Q(x)yⁿ. Uses the substitution v = y¹⁻ⁿ to reduce to a linear ODE. Shows the transformed equation, integrating factor, and solution. Select from preset equations. All calculations are client-side.

Bernoulli Equation Solver Features

  • Substitution v=y¹⁻ⁿ
  • Linear reduction
  • Integrating factor
  • Presets
  • Solution
Bernoulli equation: y' + P(x)y = Q(x)yⁿ (n ≠ 0,1). Substitute v = y¹⁻ⁿ, then v' + (1−n)P(x)v = (1−n)Q(x). This is linear in v — solve with integrating factor μ = e^∫(1−n)P dx. Then y = v^(1/(1−n)).

How to Use

Select an equation:

  • P(x), Q(x), n: Bernoulli parameters
  • Substitution: v = y¹⁻ⁿ
  • Solution: y(x)

Method

  1. Identify P(x), Q(x), n
  2. Set v = y¹⁻ⁿ
  3. v' = (1−n)y⁻ⁿy'
  4. Get linear ODE in v
  5. Solve, back-substitute

Special Cases

  • n=0: already linear
  • n=1: separable (exponential)
  • n=2: Riccati connection
  • n=3: common in applications

Step-by-Step Instructions

  1. 1Select equation.
  2. 2View substitution.
  3. 3Get linear form.
  4. 4Solve for v.
  5. 5Back-substitute for y.

Bernoulli Equation Solver — Frequently Asked Questions

Why does the substitution work?+

Dividing by yⁿ gives y⁻ⁿy' + Py¹⁻ⁿ = Q. Since d/dx[y¹⁻ⁿ] = (1−n)y⁻ⁿy', we get v'/(1−n) + Pv = Q, which is linear in v. The key insight: a power substitution linearizes the equation.

What if n is negative?+

The method still works! v = y¹⁻ⁿ = y¹⁺|ⁿ|. The linear ODE in v has different coefficients, but the same structure. Just be careful with the domain (y must not be zero if n < 0).

How is Bernoulli related to Riccati equations?+

Riccati: y' = P(x) + Q(x)y + R(x)y². This is Bernoulli when P(x) = 0. If you know one particular solution y₁ of a Riccati equation, the substitution y = y₁ + 1/v reduces it to a Bernoulli equation.

More Calculators Tools

Explore all 639 other calculators tools available on Generatr

1-Sample Z-Test Calculator2-Sample Z-Test Calculator3D Surface Area CalculatorA/B Testing Significance CalculatorAbsolute Value CalculatorAbundant Number CheckerAccelerated Aging CalculatorAchromatic Number CalculatorAcyclic Chromatic Number CalculatorAdditive Persistence CalculatorAge CalculatorAging Test CalculatorAlbertson Index CalculatorAlgebraic Connectivity CalculatorAliquot Sequence CalculatorAliquot Sum CalculatorAlternating Series Test CalculatorAmicable Number FinderAnnuity CalculatorAPY CalculatorAquarium CO2 Drop CheckerArea CalculatorArea Under Curve CalculatorArithmetic Sequence CalculatorArmstrong Number CalculatorASCVD Risk CalculatorAspect Ratio CalculatorAtom Bond Connectivity CalculatorAugmented Eccentric Connectivity CalculatorAugmented Zagreb Index Calculator
View all 639 calculators tools
Share this tool: