Linearization of a Riccati equation
= Linearization of a Riccati equation
{title2=$y=x\prime/(cx)$}
For constant nonzero $c$, putting $y=x'/(cx)$ in $y'+cy^2+a(t)y+b(t)=0$ cancels the quadratic derivative terms and gives $x''+a(t)x'+cb(t)x=0$. The <logarithmic derivative> is defined on intervals where $x\ne0$, and multiplying $x$ by a nonzero scalar does not change $y$. For variable $c(t)\ne0$, the derivative coefficient becomes $a-c'/c$.