if the quadratic term in a Ricatti equation q2 is nonzero, y'=q0+q1+q2y^2 then you can make a substitution v=yq2, S=q2q0, R=q1+(q2'/q2) which satisfies a Riccati, v'=v^2+R(x)v+S(x)=(yq2)'=q0q2+(q1+q2'/q2)v+v^2

with double substitution v=-u'/u, u now satisfies a linear 2nd ODE:

u''-R(x)u'+S(x)u=0, v'=-(u'/u)'=-(u''/u)+v^2, u''/u=v^2-v'=-S+Ru'/u, therefore we have reduced the Ricatti to an equation

u''+Su-Ru'=0, solving this equation will lead to a solution y=-u'/(q2u).....what was the original motivation behind the invention of the substitutions? my eyes are so bad I'm refonting