Reduction to Riccati by giving the symmetry to dsolve
The reduced ODE above is of Riccati type
Converting this ODE into a first order ODE of Riccati type
In the answer returned by convert, there are the Riccati ODE and the transformation of the variable used. Changes of variables in ODEs can be performed using ?PDEtools[dchange]. For example, using the transformation of variables above, we can recover the result returned by convert.