You CAN Ace Differential Equations

Topics You Need To Understand For This Page

Related Topics and Links

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

how to read math books

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

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 solution video

video by PatrickJMT

close solution

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

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 solution video

video by PatrickJMT

2418 solution video

video by Engineer4Free

close solution

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

Problem Statement

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

Solution

2419 solution video

video by MIP4U

close solution

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

Problem Statement

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

Solution

2420 solution video

video by MIP4U

close solution

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

Problem Statement

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

Solution

2421 solution video

close solution

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

Problem Statement

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

Solution

2422 solution video

video by blackpenredpen

close solution

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

Problem Statement

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

Solution

2423 solution video

close solution

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

Problem Statement

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

Solution

2424 solution video

video by MIP4U

close solution

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

Problem Statement

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

Solution

2425 solution video

video by blackpenredpen

close solution

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

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 solution video

2426 solution video

2426 solution video

video by MIP4U

close solution

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

Problem Statement

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

Solution

2427 solution video

close solution

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

Problem Statement

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

Solution

2428 solution video

video by Engineer4Free

close solution

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

Problem Statement

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

Solution

2429 solution video

video by MIP4U

close solution
Real Time Web Analytics