## 17Calculus Differential Equations - Bernoulli Equation

##### 17Calculus

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

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.

### 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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