Link to actual problem [7807] \[ \boxed {\left (x^{2}+2\right ) y^{\prime \prime }+3 x y^{\prime }-y=0} \]
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 {x^{2}+2}+x \right )^{\sqrt {2}}}{\sqrt {x^{2}+2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x^{2}+2}\, \left (\sqrt {x^{2}+2}+x \right )^{-\sqrt {2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (\sqrt {x^{2}+2}+x \right )^{-\sqrt {2}}}{\sqrt {x^{2}+2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x^{2}+2}\, \left (\sqrt {x^{2}+2}+x \right )^{\sqrt {2}} y\right ] \\ \end{align*}