Link to actual problem [9576] \[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{2}+c x +d \right ) y=0} \]
type detected by program
{"unknown"}
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 \sqrt {-b}} \operatorname {HeunC}\left (4 \sqrt {-b}, -1+\frac {a}{2}, -1+\frac {a}{2}, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x \sqrt {-b}} y}{\operatorname {HeunC}\left (4 \sqrt {-b}, -1+\frac {a}{2}, -1+\frac {a}{2}, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{x \sqrt {-b}} \left (\frac {x}{2}-\frac {1}{2}\right )^{\frac {a}{4}} \left (\frac {x}{2}+\frac {1}{2}\right )^{1-\frac {a}{4}} \left (x^{2}-1\right )^{-\frac {a}{4}} \operatorname {HeunC}\left (4 \sqrt {-b}, 1-\frac {a}{2}, -1+\frac {a}{2}, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 \,{\mathrm e}^{-x \sqrt {-b}} \left (\frac {x}{2}-\frac {1}{2}\right )^{-\frac {a}{4}} \left (\frac {x}{2}+\frac {1}{2}\right )^{\frac {a}{4}} \left (x^{2}-1\right )^{\frac {a}{4}} y}{\left (1+x \right ) \operatorname {HeunC}\left (4 \sqrt {-b}, 1-\frac {a}{2}, -1+\frac {a}{2}, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}