Maple solves this first order ode correctly and solution verifies OK. The solution is explicit and has RootOf. I asked it to solve it using dAlembert method to compare with my own solution using dAlembert.
Next I called dsolve on same ode and IC but asked for implicit solution now instead, then called solve on the implicit solution to see if it will give same solution y(x) as before (with RootOf). But instead solve gives this internal error
Error, (in evala) reducible RootOf detected.
Next called PDEtools:Solve to see if it will give same error. It did not give error. But it also could not solve for y(x) either.
Any one has any idea why this error is generated by solve? Is this expected or not?
I was expecting same result as first call to dsolve which returned explicit solution. And why is PDEtools:Solve do not generate same error? It must have run through different code path.
May be some one have some insight on this.
> 
ode:=(2*x+y(x))+(4*x2*y(x)+1)*diff(y(x),x)=0;
IC:=y(1/2)=0;

> 
maple_sol:=dsolve([ode,IC],[dAlembert]);

> 
odetest(maple_sol,[ode,IC])

> 
maple_sol:=dsolve([ode,IC],[dAlembert],implicit);

> 
#solve gives error
sol:=solve(maple_sol,y(x));

Error, (in evala) reducible RootOf detected. Substitutions are {RootOf(6*RootOf(_Z^241,index = 1)*_Z+41*_Z^2+5,label = exptmp) = 5/41*RootOf(_Z^241,index = 1), RootOf(6*RootOf(_Z^241,index = 1)*_Z+41*_Z^2+5,label = exptmp) = 1/41*RootOf(_Z^241,index = 1)}
> 
#PDEtools:Solve gives no error but does not solve it
sol:=PDEtools:Solve(maple_sol,y(x));


Download reducible_rootof_detected_august_24_2024.mw