Link to actual problem [9590] \[ \boxed {\left (x +1\right )^{2} y^{\prime \prime }+\left (x^{2}+x -1\right ) y^{\prime }-\left (x +2\right ) y=0} \]
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 &= {\mathrm e}^{-x} \left (1+x \right ) \operatorname {HeunD}\left (4, 4, -8, 12, \frac {x}{2+x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{\left (1+x \right ) \operatorname {HeunD}\left (4, 4, -8, 12, \frac {x}{2+x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (1+x \right ) \operatorname {HeunD}\left (-4, 4, -8, 12, \frac {x}{2+x}\right ) {\mathrm e}^{\frac {-1+x}{2 x +2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {-1+x}{2 x +2}} y}{\left (1+x \right ) \operatorname {HeunD}\left (-4, 4, -8, 12, \frac {x}{2+x}\right )}\right ] \\ \end{align*}