Solving a Non-linear Differential Equation: Help Needed

In summary, the conversation discusses a differential equation that needs to be solved for the general solution. The equation is non-linear and cannot be easily rearranged or separated. The person asking for help tried differentiating both sides to make the equation more manageable, but was unsuccessful. Another person mentions that there is no nice solution and only several Bessel functions are involved. The original person was expecting a nice solution for the general solution and a phase diagram, but is now aware that it may not be possible.
  • #1
Benny
584
0
Hi, can someone please help me with the following differential equation? I need to find the general solution.

[tex]
x\frac{{dy}}{{dx}} = x^2 - y^2
[/tex]

It's non-linear so I didn't bother with rearranging the equation. It doesn't look seperable either so that doesn't really leave me with much to go on with the knowledge that I have. Since the basic techniques were not applicable I tried differentiating both sides wrtx and other things like that to see if I could get the equation into a form which is easier to work with. That didn't get me anywhere so could someone help me out?

There might be something simple that I'm missing, after all it took me a few days to remember that y' = (y)^2 is solvable by separation of variables (:biggrin:) so even a small suggestion would be helpful.
 
Physics news on Phys.org
  • #2
This does not have a nice solution. What do you need this for? It's an inexact equation with no single-variable integrating factor. The solutions Maple finds are in terms of several Bessel functions.
 
  • #3
I thought that there would be a fairly neat solution. The question asks for the exact general solution and then asks for a phase diagram of some system so I expected a 'nice' answer. Thanks anyway.
 

FAQ: Solving a Non-linear Differential Equation: Help Needed

What is a non-linear differential equation?

A non-linear differential equation is a mathematical equation that involves one or more derivatives of a dependent variable with respect to one or more independent variables, and the dependent variable is not proportional to the independent variable. This means that the rate of change of the dependent variable is not directly proportional to the independent variable.

Why is it important to solve non-linear differential equations?

Non-linear differential equations are used to model a wide range of complex systems in science, engineering, and economics. By solving these equations, we can gain a better understanding of how these systems behave and make predictions about their future behavior. Non-linear differential equations also have many real-world applications, such as in weather forecasting, chemical reactions, and population dynamics.

What are the methods for solving non-linear differential equations?

There are several methods for solving non-linear differential equations, including separation of variables, substitution, integrating factors, and numerical methods. These methods can be used to find an exact solution or an approximate solution to the equation. The choice of method depends on the complexity of the equation and the desired level of accuracy.

How do I know if my solution to a non-linear differential equation is correct?

To check if your solution to a non-linear differential equation is correct, you can substitute it into the original equation and see if it satisfies the equation. You can also use numerical methods to approximate the solution and compare it to your analytical solution. Additionally, you can use software programs or online calculators that can solve non-linear differential equations and verify your solution.

Are there any tips for solving non-linear differential equations?

Some tips for solving non-linear differential equations include:

  • Identify the type of equation and choose an appropriate method for solving it
  • Try to reduce the equation to a simpler form by using substitutions or transformations
  • Check for symmetry and use it to simplify the equation
  • Make sure to check your solution for correctness
  • Use software programs or online calculators for complex equations

Similar threads

Linearly independent functions with identically zero Wronskian
Replies
1
Views
917
Solving system of differential equations using elimination method
Replies
10
Views
899
Solve the given differential equation
Replies
4
Views
1K
Solve the system of differential equations
Replies
10
Views
1K
Solution of a parametric differential equation
Replies
4
Views
2K
Given unit impulse response, obtain differential equation
Replies
7
Views
976
Solving a second-order differential equation
Replies
33
Views
3K
Understanding Wronskian of solutions to homoeneous 2nd order linear DE
Replies
2
Views
777
Find the differential equation
Replies
7
Views
1K
Finding the differential equation of motion
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top