Infinite series solution for NON-linear ODEs?

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The infinite series method, specifically Frobenius' method, is generally not applicable for obtaining general solutions of non-linear ordinary differential equations (ODEs) due to the additive nature of solutions that only holds for linear equations. However, non-linear ODEs can be approached using power series expansions, though specific references for this method are scarce. For first-order differential equations, the power series method may be viable due to the existence and uniqueness theorem. For higher-order non-linear ODEs, the Adomian method is suggested as an iterative approach that can yield quickly converging series solutions. Overall, while challenges exist in solving non-linear ODEs with series methods, alternative approaches like the Adomian method may provide viable solutions.
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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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No. Frobenius' method and series methods in general assume that you can "add" solutions. That is only true for linear differential equations.
 


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?
 


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)
 

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