Infinite series solution for NON-linear ODEs?

In summary, the conversation explores the possibility of using the infinite series method (Frobenius) to solve non-linear differential equations. The speaker suggests that while this method works for linear equations, it may not be applicable for non-linear equations. They also mention the Adomian method as a possible alternative for obtaining series solutions for higher order non-linear ODEs. The speaker also expresses a desire to find a resource specifically focused on solving non-linear ODEs using series methods.
  • #1
smallphi
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infinite series solution for NON-linear ODEs?

Is it possible to use the infinite series method (Frobenius) to obtain general solutions of non-linear ODE's, I want to try a second order equation. Any good references where I can see how that goes exactly?
 
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  • #2


No. Frobenius' method and series methods in general assume that you can "add" solutions. That is only true for linear differential equations.
 
  • #3


The solution of a non-linear ODE is a function that can be expanded in power series and I've actually seen non-linear ODE's solved that way. The question is wether there is a book/article that focuses specifically on such type of solving cause I don't want to spend months reinventing the wheel and the hot water?
 
  • #4


May be for first order DE it is possible to use power series method because of the existence and uniqueness theorem.

For higher order, if you are still interested in series solution, try the Adomian method. I understand that it is an iterative method but the series obtained converges quickly (please forgive me if I'm wrong. I only saw it in seminars. Hopefully I will be able to learn properly this method one day)
 

FAQ: Infinite series solution for NON-linear ODEs?

1. What is an infinite series solution for NON-linear ODEs?

An infinite series solution for NON-linear ODEs is a method used to approximate the solution of a non-linear ordinary differential equation (ODE) by expressing it as an infinite sum of terms. This is often used when the ODE cannot be solved analytically using traditional methods.

2. How does an infinite series solution differ from other methods of solving NON-linear ODEs?

An infinite series solution differs from other methods of solving NON-linear ODEs in that it involves expressing the solution as an infinite sum of terms, rather than a single equation or function. This allows for a more precise and accurate approximation of the solution.

3. What are the benefits of using an infinite series solution for NON-linear ODEs?

One of the main benefits of using an infinite series solution for NON-linear ODEs is that it allows for a more accurate and precise approximation of the solution compared to other methods. It also allows for the solution to be expressed in a simpler form, making it easier to work with and manipulate.

4. Are there any limitations to using an infinite series solution for NON-linear ODEs?

Yes, there are limitations to using an infinite series solution for NON-linear ODEs. One limitation is that the series may not converge for certain values of the independent variable, making the solution invalid. Additionally, the series may become increasingly complex and difficult to work with as more terms are added, making it less practical for certain problems.

5. Can an infinite series solution be used for any type of non-linear ODE?

No, an infinite series solution may not be applicable for all types of non-linear ODEs. It is most commonly used for ODEs with polynomial or trigonometric functions, but may not be suitable for ODEs with more complex functions or boundary conditions.

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