Link to actual problem [10618] \[ \boxed {x y^{\prime }-f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}=-a} \]
type detected by program
{"riccati"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{f \left (x \right )}, \underline {\hspace {1.25 ex}}\eta &= -\frac {a}{f \left (x \right )}\right ] \\ \left [R &= y+a \ln \left (x \right ), S \left (R \right ) &= \int \frac {f \left (x \right )}{x}d x\right ] \\ \end{align*}