Link to actual problem [9521] \[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y=0} \]
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 &= \sqrt {x}\, {\mathrm e}^{-x} \operatorname {HeunD}\left (4, 3, -8, 5, \frac {-1+x}{1+x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{\sqrt {x}\, \operatorname {HeunD}\left (4, 3, -8, 5, \frac {-1+x}{1+x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, {\mathrm e}^{-\frac {1}{x}} \operatorname {HeunD}\left (-4, 3, -8, 5, \frac {-1+x}{1+x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {1}{x}} y}{\sqrt {x}\, \operatorname {HeunD}\left (-4, 3, -8, 5, \frac {-1+x}{1+x}\right )}\right ] \\ \end{align*}