The formula given in this answer to transform a Riccati ODE $y'(x)=p(x)+q(x)y(x)+r(x)(y(x))^2$ into a second-order linear ODE is $u(x)=-\frac{y'(x)}{r(x)y(x)}$. However, for an example such as $y' = -\frac{3}{t^2} + \frac{1}{t}y + y^2$, it isn't clear to me that the substitution $u = -\frac{y'}{y}$ eliminates all references to $y$ from the ODE. In chat, the substitution $y = -\frac{u'}{u}$ was suggested for this example, which does work.
My question is what the general formula is for this substitution, in cases where the coefficient of $y^2$ isn't necessarily $1$. Interpreting the linked formula as a typo and pattern matching might suggest the answer to be $y = -\frac{u'(x)}{r(x)u(x)}$. Is this the correct forumla, and if not, what should it be?