First order nonlinear ordinary differential equation

In summary, the given differential equation, y' + Ay2 = B, where A and B are constants and y is a function of x, can be solved by rearranging it to get dy/dx = B-Ay^2, which is a separable equation that can be solved using integration. After solving the integrals, the solution for y in terms of x can be obtained.
  • #1
MadMathMan
2
0

Homework Statement



y' + Ay2 = B

A & B are constants and y is a function of x

Find the general solution to the differential equation. (Find y(x)).

Homework Equations





The Attempt at a Solution



This differential equation came up when I was trying to solve a problem in physics. I have just learned to solve basic linear differential equations from high school and don't know how to start solving this or even if it's solvable without a computer. If somebody could give me a push in the right direction or tell me what I could study to be able to solve it, it would have been very nice :)

The only thing I've tried is differentiating it with respect to x, and getting this:

y'' + 2Ayy' = 0

I don't find this any easier because I have the y×y' there. So, any help would be nice ;)
 
Physics news on Phys.org
  • #2
Write the ODe in the following way:
[tex]
\frac{dy}{dx}=B-Ay^{2}
[/tex]
This equation is separable, divide by [tex]B-Ay^{2}[/tex] and integrate

Mat
 
  • #3
This one is "separable" - which means it's solvable using integration. Rearrange to get
[tex]
\frac{dy}{B-Ay^2} = dx,
[/tex]
and do two integrals. There will be significant algebraic rearrangement involved to solve for y in terms of x.

You should probably reference a differential equations book.
 
  • #4
Thanks for the answers!
That was really less painful than I expected :tongue: Don't know why I didn't think of that :uhh:
Next problem now is solving the integral, but I think I should be able to do that myself
 

What is a first order nonlinear ordinary differential equation?

A first order nonlinear ordinary differential equation is a mathematical equation that relates an unknown function to its derivative, where the function and its derivative are both nonlinear. It is a type of differential equation that is commonly used to describe the behavior of dynamic systems in various scientific fields.

What is the difference between a linear and nonlinear differential equation?

A linear differential equation has a linear relationship between the unknown function and its derivative, meaning the function and its derivative are raised to the first power. In contrast, a nonlinear differential equation has a non-linear relationship between the unknown function and its derivative, meaning the function and its derivative can be raised to any power.

How do you solve a first order nonlinear ordinary differential equation?

There is no one method for solving all first order nonlinear ordinary differential equations, as the approach can vary depending on the specific equation. However, some common techniques include separation of variables, integrating factors, and substitution methods. In some cases, it may also be necessary to use numerical methods to approximate a solution.

What are some real-world applications of first order nonlinear ordinary differential equations?

First order nonlinear ordinary differential equations have a wide range of applications in various fields of science and engineering. Some examples include modeling population growth, analyzing chemical reactions, predicting the spread of diseases, and understanding the behavior of electrical circuits.

What are the limitations of using first order nonlinear ordinary differential equations?

First order nonlinear ordinary differential equations can become very complex and difficult to solve for certain systems, especially when they involve multiple variables and parameters. In addition, they may not always accurately represent real-world systems due to the simplifications and assumptions made in the mathematical models. Furthermore, the solutions obtained may only be valid within certain ranges of the input variables and may not accurately predict the behavior of the system outside of these ranges.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
409
  • Calculus and Beyond Homework Help
Replies
7
Views
641
  • Calculus and Beyond Homework Help
Replies
7
Views
133
Replies
7
Views
465
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
221
  • Calculus and Beyond Homework Help
Replies
5
Views
846
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
877
  • Calculus and Beyond Homework Help
Replies
33
Views
3K
Back
Top