Help with bernoulli differential equation

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Start date Mar 29, 2012
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Bernoulli Differential Differential equation

In summary, the Bernoulli differential equation is a first-order nonlinear differential equation used in physics and engineering, named after mathematician Daniel Bernoulli. Its general solution is given by y(x) = (1/n) * (y^-n + C), obtained by using the substitution v = y^(1-n). The equation has various applications and can be solved by recognizing its form, using a substitution, solving a linear equation, and using an inverse substitution. Its special cases, n = 0 and n = 1, require different methods for solving.

Mar 29, 2012

AoD

I need help with solving equation y'=(x^2-y^2)/xy as bernoulli differential equation.

I also tried to do it differently by plugging in y=xv, so I got f(v)=1/v-v.

Finally I got answer y=(+/-) [(x^4+c)^(1/2)]/[(x*2^(1/2)] but I still have no idea how to do it as bernoulli equation

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Mar 29, 2012

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Related to Help with bernoulli differential equation

1. What is the Bernoulli differential equation?

The Bernoulli differential equation is a type of first-order nonlinear differential equation that can be written in the form dy/dx + P(x)y = Q(x)y^n, where n is any real number except 0 and 1. It is named after the Swiss mathematician Daniel Bernoulli who first studied it.

2. What is the general solution to the Bernoulli differential equation?

The general solution to the Bernoulli differential equation is given by the formula y(x) = (1/n) * (y^-n + C), where C is a constant. This solution can be obtained by using the substitution v = y^(1-n) and solving the resulting linear differential equation.

3. How is the Bernoulli differential equation useful in physics and engineering?

The Bernoulli differential equation has many applications in physics and engineering, particularly in fluid dynamics. It can be used to model various phenomena such as the flow of fluids through pipes, the lift on an airplane wing, and the motion of projectiles. It is also useful in analyzing economic and biological systems.

4. What are the key steps in solving a Bernoulli differential equation?

The key steps in solving a Bernoulli differential equation are: 1) recognizing the form of the equation, 2) using the substitution v = y^(1-n) to transform it into a linear differential equation, 3) solving the resulting linear equation using standard techniques, and 4) using the inverse substitution to obtain the final solution.

5. Are there any special cases of the Bernoulli differential equation?

Yes, there are two special cases of the Bernoulli differential equation: n = 0 and n = 1. When n = 0, the equation reduces to a linear differential equation, and when n = 1, it becomes a separable differential equation. These cases have their own specific methods for solving them, and the general solution formulas will not apply.