Link to actual problem [7632] \[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+1\right ) y^{\prime }+y=0} \]
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 {x}{\sqrt {x^{2}+1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x^{2}+1}\, y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )}{\sqrt {x^{2}+1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x^{2}+1}\, y}{x \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )}\right ] \\ \end{align*}