Hello:   
      
   In my research on polynomials, we see variety of differential equations   
   representing each polynomial.   
      
   Well, if we get a fixed pattern of solutions by Galois methods, here we have   
   many different forms (using differential equations).   
      
   Particularly solutions are different than those obtained by Galois Theory.   
      
   The fundamental theorem of algebra (FTA) telling us that we have at most n   
   distinct solutions for polynomial of degree n.   
      
   Here we have opportunity to test the FTA, particularly with higher order   
   polynomials, where from Galois Theory point of view it is more difficult to   
   approach.   
      
   In my other volumes of my books, I hope to address this issue with many   
   examples.   
      
   So, your question is difficult one (from my point of view), depending on who   
   is the researcher: (using Galois Theory or differential equations to study the   
   matter).   
      
   In my books, Galois Theory will be a diminished concept in the decades ahead.   
      
   It is even difficult to say to current algebraists that we even need to test   
   FTA (i.e. the theorem should be more under scrutiny) within hundreds of   
   different solutions, obtained by differential equations methods.   
      
   It is also not easy, if algebraists realize that their current research (on   
   modern Algebra) will be not much of use in the future.   
      
   I hope I have answered your question.   
      
      
   Sincerely   
      
   Dr.Mehran Basti   
      
      
   PS: In Volume 1, we learn basics of solving polynomials as a class with   
   differential equations (In print by the end of December 2015).   
      
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