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Expert ∞ Differential Equations, Abstract Algebra & Applied Math Topic 11 of 97

Bernoulli Equations

Substitute v = y¹⁻ⁿ to linearise.

Key idea

Substitute v = y¹⁻ⁿ to linearise.

In depth

y′ + P y = Q yⁿ is nonlinear but becomes linear with the substitution v = y^(1−n). Then solve with an integrating factor and substitute back.

Worked example

y′ + y = y²: let v = 1/y, so v′ − v = −1, v = 1 + Ce^t, y = 1/(1 + Ce^t).

Formula

y′ + py = qyⁿ

Real-world uses

Common mistake

Forgetting the solution y = 0.

How to learn this

  1. Read the key idea and say it in your own words.
  2. Work through the worked example, then try a similar problem.
  3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

What is Bernoulli Equations?

Substitute v = y¹⁻ⁿ to linearise. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.

What is the formula for Bernoulli Equations?

The key formula is y′ + py = qyⁿ.

Where is Bernoulli Equations used in real life?

Real-world examples: Modelling population growth with harvesting and some chemical reaction rates.

What should I learn before Bernoulli Equations?

Make sure you are comfortable with Exact Equations, the previous topic in Differential Equations, Abstract Algebra & Applied Math.

What is a common mistake with Bernoulli Equations?

Forgetting the solution y = 0.

How can I practise Bernoulli Equations?

Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.

What comes after Bernoulli Equations?

Move on to Euler’s Method next.

Related expert topics in Differential Equations, Abstract Algebra & Applied Math