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It seems that the non-linear differential equation satisfied by the Lambert $W$ function is simple enough for this question to have already been answered. This paper proves that the Lambert $W$ is non-elementary by appealing to a result of Rosenlicht (1969): Bronstein, M., Corless, R. M., Davenport, J. H. and Jeffrey, D. J. (2008) Algebraic properties of ...


11

In [Ritt 1948], the method of J. Liouville is given for the Kepler equation. [Ritt 1948] Ritt, J. F.: Integration in finite terms. Liouville's theory of elementary methods. 1948, page 56 A further method is the method of Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22. It ...


4

Your belief is not correct, as Robert Israel pointed in his comment. Same for differential equations: $yy'-(3/2)z^2=0$ has a solution $y(z)=z^{3/2}$ which is not analytic at $0$. The correct theorem is: If $P(z_0,y_0,y_1,\ldots,y_{n})=0$, where $P$ is a polynomial, (or an analytic function in a neighborhood of $(z_0,\ldots,y_n)$), and $\partial P/\partial y_{...


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It is not true. See the example in my paper Algebraic D-groups and differential galois theory, Pacific Journal Math, vol 216, No. 2, 2004. It is discussed on p. 356.


3

[This should rather be a comment, but I have not yet enough reputation for comments, so I post it as an answer.] I do not have access now to the book you are citing and therefore cannot properly compare the definitions, but Newton polygons are used in the classification of linear differential equations. You can an introduction in 3.3 of M. van der Put and M....


2

An analytic solution to the Abel equation of the second kind is claimed in this 2015 preprint by Rostami. As they say on Twitter, sharing does not constitute endorsement, but do check it out.


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