This ODE is extremely unlikely to have an explicit solution. Mathematica cannot do anything with this ODE even when $g(x)\equiv x$. 

Here is an image of the corresponding Mathematica notebook: 

[![enter image description here][1]][1] 

And here is [a link][2] to the Mathematica notebook itself. 


  [1]: https://i.sstatic.net/Soz2V.png
  [2]: https://u.pcloud.link/publink/show?code=XZA40TVZCAyy3X73DjYvXzl1ySX8IybILJ07