Solving a Bernoulli differential equation

In summary, the person is seeking help with a differential equation that they have been trying to solve using a specific method. They have made a mistake and corrected it, but the book and Wolfram have different answers. They are wondering how the simplification was done in Wolfram's answer and where the problem may lie after the correction.
  • #1
Boxiom
7
0
Hello!

I'm stuck at the moment with this differential equation. I've been trying to use the method for solving these equations, but my answer is not correct according to my book. Could anyone please explain what I'm doing wrong? Thanks!

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  • #2
Hi !
see attachment
 

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  • #3
Hi and thanks for your reply =)

A stupid mistake from my side, I changed it up and the only difference in the answer is that te^-t is now -te^-t. Still, the book doesn't agree with me unless they have the answer on some weird form. I also checked wolfram, they gave me this answer:

WJ6eFxI.gif


What exactly did they do to simplify it like that?
 
  • #4
Where is the problem after the correction of sign ?
 

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  • #5


Hello there,

Solving a Bernoulli differential equation can be a challenging task, but with the right approach, it can be achieved. Before we dive into your specific question, let's review the general method for solving Bernoulli differential equations.

First, it is important to note that Bernoulli differential equations are nonlinear, meaning they cannot be solved using the standard methods for linear differential equations. The key to solving a Bernoulli equation is to transform it into a linear equation by using a substitution. This substitution typically involves dividing both sides of the equation by the highest power of the dependent variable, which in this case is usually y.

Once the substitution is made, the equation can be solved using standard methods for linear differential equations, such as separation of variables or integrating factors. After solving for the dependent variable, the original substitution must be reversed to find the solution for the original equation.

Now, let's address your specific question. Without knowing the exact equation and your attempted solution, it is difficult to pinpoint where you may have made a mistake. However, some common errors in solving Bernoulli equations include incorrect substitutions, mistakes in algebraic manipulations, and errors in integrating factors.

I would suggest double-checking your work and making sure you have followed the correct steps for solving a Bernoulli equation. If you are still having trouble, it may be helpful to seek assistance from a tutor or your professor.

I hope this helps and good luck with your problem-solving!
 

What is a Bernoulli differential equation?

A Bernoulli differential equation is a type of differential equation that can be written in the form of y' + P(x)y = Q(x)yn, where n is a constant. It is named after the Swiss mathematician Jacob Bernoulli who first studied this type of equation in the 18th century.

What is the general method for solving a Bernoulli differential equation?

The general method for solving a Bernoulli differential equation involves using a substitution v(x) = y1-n to transform the equation into a linear differential equation. This can then be solved using standard techniques such as separation of variables or integrating factors.

Can all Bernoulli differential equations be solved analytically?

No, not all Bernoulli differential equations can be solved analytically. Some may require numerical methods or approximation techniques to find a solution. However, there are certain special cases of Bernoulli differential equations that can be solved analytically, such as when n is equal to 0, 1, or 2.

What are the applications of solving Bernoulli differential equations?

Bernoulli differential equations have many applications in various fields of science and engineering. They are commonly used to model growth and decay processes, as well as in fluid dynamics, population dynamics, and economics. They also have applications in control theory and electrical circuits.

Is there a specific technique for solving nonlinear Bernoulli differential equations?

Yes, there is a special technique called the Riccati substitution that can be used to solve nonlinear Bernoulli differential equations. This involves using a substitution v(x) = y' + P(x)y to transform the equation into a linear differential equation, which can then be solved using the standard methods mentioned earlier.

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