## 17Calculus Differential Equations - Bernoulli Equation

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

video by MIT OCW

### Practice

Instructions - - 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

### 2417 video

video by PatrickJMT

$$t^2y'+2ty-y^3=0$$

Problem Statement

Solve the Bernoulli Equation $$t^2y'+2ty-y^3=0$$.

Solution

We found two solutions to this problem presented by two different people. If the first one doesn't help you, try the second one.

### 2418 video

video by PatrickJMT

### 2418 video

video by Engineer4Free

$$y'+xy=xy^2$$

Problem Statement

Solve the Bernoulli Equation $$y'+xy=xy^2$$.

Solution

### 2419 video

video by MIP4U

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

Problem Statement

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

Solution

### 2420 video

video by MIP4U

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

Problem Statement

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

Solution

### 2421 video

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

Problem Statement

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

Solution

### 2422 video

video by blackpenredpen

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

Problem Statement

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

Solution

### 2423 video

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

Problem Statement

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

Solution

### 2424 video

video by MIP4U

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

Problem Statement

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

Solution

### 2425 video

video by blackpenredpen

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

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 in case one helps you more than the other.

### 2426 video

video by MIP4U

$$y' + 2xy = xy^3$$

Problem Statement

Solve the Bernoulli Equation $$y' + 2xy = xy^3$$.

Solution

### 2427 video

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

Problem Statement

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

Solution

### 2428 video

video by Engineer4Free

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

Problem Statement

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

Solution

### 2429 video

video by MIP4U

You CAN Ace Differential Equations

 integrating factors - linear

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