Link to actual problem [11001] \[ \boxed {\left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+3 \left (a x +b \right ) y^{\prime }+d y=0} \]
type detected by program
{"kovacic"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\left (\sqrt {a \left (x^{2} a +2 b x +c \right )}+x a +b \right )}^{\frac {\sqrt {-d +a}}{\sqrt {a}}}}{\sqrt {x^{2} a +2 b x +c}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x^{2} a +2 b x +c}\, {\left (\sqrt {a \left (x^{2} a +2 b x +c \right )}+x a +b \right )}^{-\frac {\sqrt {-d +a}}{\sqrt {a}}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\left (\sqrt {a \left (x^{2} a +2 b x +c \right )}+x a +b \right )}^{-\frac {\sqrt {-d +a}}{\sqrt {a}}}}{\sqrt {x^{2} a +2 b x +c}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x^{2} a +2 b x +c}\, {\left (\sqrt {a \left (x^{2} a +2 b x +c \right )}+x a +b \right )}^{\frac {\sqrt {-d +a}}{\sqrt {a}}} y\right ] \\ \end{align*}