Link to actual problem [10965] \[ \boxed {x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y=0} \]
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 &= x^{-\frac {c}{2}+\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}} {\mathrm e}^{\frac {x \left (-x \,a^{2}-2 a b +2 A \right )}{2 a}} \operatorname {HeunB}\left (\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {c}{2}} x^{-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} {\mathrm e}^{-\frac {x \left (-x \,a^{2}-2 a b +2 A \right )}{2 a}} y}{\sqrt {x}\, \operatorname {HeunB}\left (\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, \frac {\left (-c -1\right ) a^{3}+2 B \,a^{2}-2 A a b +2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {c}{2}-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}+\frac {1}{2}} {\mathrm e}^{\frac {x \left (-x \,a^{2}-2 a b +2 A \right )}{2 a}} \operatorname {HeunB}\left (-\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {c}{2}} x^{\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} {\mathrm e}^{-\frac {x \left (-x \,a^{2}-2 a b +2 A \right )}{2 a}} y}{\sqrt {x}\, \operatorname {HeunB}\left (-\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, \frac {\left (-c -1\right ) a^{3}+2 B \,a^{2}-2 A a b +2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )}\right ] \\ \end{align*}