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I would like an example with proof or appropriate reference of a differential equation that does not admit analytic (in the sense of convergent power series, not in the sense of closed form) solutions, excepting perhaps some thin set.

I do know of a paper on a differential equation with no computable solutions, and since analytic would imply computable, this counts for the non analytic part . But, I would rather an example, or by preference a broad family of examples, in which the the solutions are computable just not analytic.

Ponder Stibbons
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    Take a differential equation whose only solution is analytic (and non-zero) and do the change of variable $y = g Y$ where $g$ is nowhere analytic. If you meant an equation with analytic coefficients then in one variable it will always have a locally analytic solution, so it remains to check the case of PDE in several variables with analytic coefficients. – reuns Jan 07 '20 at 06:29
  • So cool. That is disturbingly straight forward. That gets me into the right mind set - so I feel that you have answered my question, although I always hesitate to mark something as a full answer until there has been time for a few people to respond if they want to. – Ponder Stibbons Jan 07 '20 at 06:48

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