# Bernoulli differential equation

• tracedinair
In summary, the conversation is discussing how to solve the Bernoulli equation, which involves using the Integrating Factor method and changing variables. The suggested variable to use is u=y^(1-n), where n is the exponent of y in the equation.
tracedinair

## Homework Statement

Solve the Bernoulli equation,

y'(x) - 4y(x) = 2e^(x) * sqrt(y(x))

## Homework Equations

y' + P(x)y = Q(x)y^n - Bernoulli Eqn
e^(∫P(x) dx) - Integrating Factor

## The Attempt at a Solution

y' - 4y = 2e^(x) * y^(1/2)

Divided both sides by y^(1/2)

y'/y^(1/2) - 4y/y^(1/2) = 2e^(x)

y'/y^(1/2) - 4y^(1/2) = 2e^(x)

My problem comes when changing variables. What am I supposed to choose for 'u' (the variable I'll be changing to)? Just y^(1/2)? My text and notes aren't very clear on this.

Last edited:
Yes put u=y1/2

In general, for:

$$\frac{dy}{dx}+P(x)y=Q(x)y^n$$

put $u=y^{1-n}$

Alright, thank you for your help.

## 1. What is a Bernoulli differential equation?

A Bernoulli differential equation is a type of first-order nonlinear differential equation that can be expressed in the form dy/dx + P(x)y = Q(x)y^n, where n is a constant. It is named after Swiss mathematician Daniel Bernoulli who first studied this type of equation in the 18th century.

## 2. What is the difference between a Bernoulli differential equation and a linear differential equation?

The main difference between a Bernoulli differential equation and a linear differential equation is that the former involves a non-constant term with a power of y, while the latter does not. This makes Bernoulli equations more challenging to solve, as they require a change of variables to convert them into linear equations.

## 3. How do you solve a Bernoulli differential equation?

To solve a Bernoulli differential equation, you first need to identify the values of P(x) and Q(x). Then, you use a substitution of the form u = y^(1-n) to transform the equation into a linear form. After solving for u, you can then substitute back to find the solution for y.

## 4. What are some applications of Bernoulli differential equations?

Bernoulli differential equations are commonly used in physics to model various phenomena, such as population growth, chemical reactions, and fluid dynamics. They are also used in economics to study the relationship between supply and demand, and in biology to model population dynamics.

## 5. Are there any real-life examples of Bernoulli differential equations?

Yes, there are many real-life examples of Bernoulli differential equations. One example is the growth rate of a bacterial population, which can be modeled using the equation dy/dt = ky, where k is a constant. Another example is the relationship between the velocity and pressure of a fluid in motion, which can be described by the Bernoulli's equation.

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