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17Calculus Differential Equations - Bernoulli Equation

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Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
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For an equation of the type \(y' = p(x)y + q(x)y^n\), called a Bernoulli Equation, we can use the special substitution \(v = y^{1-n}\), which will turn the equation into a linear equation.
Note: This technique uses integrating factors in order to solve the resulting linear equation.

Here are several good videos explaining the theory of how and why this substitution works.

PatrickJMT - Bernoulli Equation for Differential Equations , Part 1 [10mins-25secs]

video by PatrickJMT

Engineer4Free - How to solve Bernoulli differential equations [5mins-29secs]

video by Engineer4Free

MIT OCW - Bernoulli Equation and example [11mins-36secs]

video by MIT OCW

Introduction to Partial Differential Equations with Applications (Dover Books on Mathematics)

Practice

Unless otherwise instructed, solve these Bernoulli Equations. Give your answers in exact form.

\(y' =(A\cos t+B)y-y^3\)

Problem Statement

Solve the Bernoulli Equation \(y' =(A\cos t+B)y-y^3\)

Solution

PatrickJMT - 2417 video solution

video by PatrickJMT

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\(t^2y'+2ty-y^3=0\)

Problem Statement

Solve the Bernoulli Equation \(t^2y'+2ty-y^3=0\)

Solution

We include two solutions to this problem by two different instructors.

PatrickJMT - 2418 video solution

video by PatrickJMT

Engineer4Free - 2418 video solution

video by Engineer4Free

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\(y'+xy=xy^2\)

Problem Statement

Solve the Bernoulli Equation \(y'+xy=xy^2\)

Solution

MIP4U - 2419 video solution

video by MIP4U

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\(\displaystyle{ xy'+y=\frac{1}{y^2} }\)

Problem Statement

Solve the Bernoulli Equation \(\displaystyle{ xy'+y=\frac{1}{y^2} }\)

Solution

MIP4U - 2420 video solution

video by MIP4U

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\(\displaystyle{ \frac{dy}{dx} + y = xy^4 }\)

Problem Statement

Solve the Bernoulli Equation \(\displaystyle{ \frac{dy}{dx} + y = xy^4 }\)

Solution

Houston Math Prep - 2421 video solution

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\(\displaystyle{ \frac{dy}{dx} = \sqrt{y} - y; y(0) = 9 }\)

Problem Statement

Solve the Bernoulli Equation \(\displaystyle{ \frac{dy}{dx} = \sqrt{y} - y; y(0) = 9 }\)

Solution

blackpenredpen - 2422 video solution

video by blackpenredpen

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\(\displaystyle{ \frac{dy}{dx} - \frac{1}{x}y = xy^2 }\)

Problem Statement

Solve the Bernoulli Equation \(\displaystyle{ \frac{dy}{dx} - \frac{1}{x}y = xy^2 }\)

Solution

Houston Math Prep - 2423 video solution

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\(\displaystyle{ y' + \frac{2}{x}y = -x^9y^5; y(1) = 1 }\)

Problem Statement

Solve the Bernoulli Equation \(\displaystyle{ y' + \frac{2}{x}y = -x^9y^5; y(1) = 1 }\)

Solution

MIP4U - 2424 video solution

video by MIP4U

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\( y' = -y(1 + xy^2) \)

Problem Statement

Solve the Bernoulli Equation \( y' = -y(1 + xy^2) \)

Solution

blackpenredpen - 2425 video solution

video by blackpenredpen

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\(\displaystyle{ y' + \frac{4}{x} y = x^3y^2 }\)

Problem Statement

Solve the Bernoulli Equation \(\displaystyle{ y' + \frac{4}{x} y = x^3y^2 }\)

Solution

This problem was solved by three different people. We include all three videos here.

The Lazy Engineer - 2426 video solution

Engineer In Training Exam - 2426 video solution

MIP4U - 2426 video solution

video by MIP4U

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\( y' + 2xy = xy^3 \)

Problem Statement

Solve the Bernoulli Equation \( y' + 2xy = xy^3 \)

Solution

Houston Math Prep - 2427 video solution

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\(\displaystyle{ y' + \frac{1}{t}y = -ty^3 }\)

Problem Statement

Solve the Bernoulli Equation \(\displaystyle{ y' + \frac{1}{t}y = -ty^3 }\)

Solution

Engineer4Free - 2428 video solution

video by Engineer4Free

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\(\displaystyle{ \frac{dy}{dt} = 2y + y^5; y(0) = 1 }\)

Problem Statement

Solve the Bernoulli Equation \(\displaystyle{ \frac{dy}{dt} = 2y + y^5; y(0) = 1 }\)

Solution

MIP4U - 2429 video solution

video by MIP4U

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Really UNDERSTAND Differential Equations

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Practice Instructions

Unless otherwise instructed, solve these Bernoulli Equations. Give your answers in exact form.

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