= Riccati cross-ratio superposition
{c}
{title2=$\frac{(f-g)(h-j)}{(f-j)(h-g)}=K$}
Three distinct particular solutions $g,h,j$ of a <Riccati equation> determine every solution by one constant <cross-ratio>. Indeed, for $f'=af^2+bf+c$, differences satisfy $(f-g)'=(a(f+g)+b)(f-g)$. The logarithmic derivatives cancel in the displayed cross-ratio, so it is constant. Solving for $f$ gives a <rational function> of the three solutions and $K$, interpreted on the <Riemann sphere> at poles.
Back to article page
