I have found another nonlinear example at sosmath.com [here][1] but it appears to contain mistakes. 

Also you can play with the examples of power series solutions in [Maple][2] using **dsolve** with the **series** option.

Some further examples could probably be found in connection with the study of the Painleve property and the [Painleve test][3] of integrability for (systems of) ODEs, see e.g. the book *Integrability and Nonintegrability of of dynamical systems* (full text available at [this URL][4]) by Alain Goriely and references therein. 

Another useful keyword to search for is *formal solution* instead of the *power series solution*, although the convergence will probably not be discussed in the context of formal solutions.


  [1]: http://www.sosmath.com/CBB/viewtopic.php?p=176220
  [2]: https://www.maplesoft.com/
  [3]: https://encyclopediaofmath.org/wiki/Painlev%C3%A9_test
  [4]: https://web.archive.org/web/20130513034003/http://math.arizona.edu/~goriely/pubB.html