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## Main Question or Discussion Point

In a Riccati equation y'=qo+q1y+q2y^2, if q2 is nonzero 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)

were these substitutions simply invented or is there a reasoning process behind the proof?

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)

were these substitutions simply invented or is there a reasoning process behind the proof?