Link to actual problem [9557] \[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+3 y^{\prime } x +a y=0} \]
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 (x +\sqrt {x^{2}+1}\right )^{\sqrt {1-a}}}{\sqrt {x^{2}+1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x^{2}+1}\, \left (x +\sqrt {x^{2}+1}\right )^{-\sqrt {1-a}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x +\sqrt {x^{2}+1}\right )^{-\sqrt {1-a}}}{\sqrt {x^{2}+1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x^{2}+1}\, \left (x +\sqrt {x^{2}+1}\right )^{\sqrt {1-a}} y\right ] \\ \end{align*}