From: edgar@math.ohio-state.edu.invalid   
      
   In Maple, the Maple solution in terms of Heun functions satisfies the   
   equation.  But the Kamke solution doesn't.  You say you got the Kamke   
   solution by hand.  So I'm guessing you wrote something wrong in your   
   post....   
      
   either the equation   
   eq:=diff(y(x),x)-y(x)^2-y(x)*sin(2*x)-cos(2*x) = 0;   
      
   or the solution   
   u:=cos(x)   
   I:=int( exp(-u^2)/u^2, x)   
   y:= tan(x) + exp(-u^2)/(u^2 * (I + C))   
      
   ..............   
      
   It is not surprising that Mathematica does not solve this, since   
   Mathematica has not yet added the Heun functions.   
      
   There is a standard way to convert a Riccati equation into a linear   
   second-order DE.  I am assuming the result can be recognized as a Heun   
   DE, and that is how Maple did its solution.   
      
   --   
   G. A. Edgar                              http://www.math.ohio-state.edu/~edgar/   
      
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