Link to actual problem [10907] \[ \boxed {x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y=0} \]
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}^{\frac {x \left (-x \,a^{2}-2 a b +2 A \right )}{2 a}} \operatorname {HeunB}\left (c -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 \operatorname {C0} \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x \left (-x \,a^{2}-2 a b +2 A \right )}{2 a}} y}{\operatorname {HeunB}\left (c -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 {\left (b c -2 \operatorname {C0} \right ) \sqrt {2}}{\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^{-c +1} {\mathrm e}^{\frac {x \left (-x \,a^{2}-2 a b +2 A \right )}{2 a}} \operatorname {HeunB}\left (-c +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 \operatorname {C0} \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{c} {\mathrm e}^{-\frac {x \left (-x \,a^{2}-2 a b +2 A \right )}{2 a}} y}{x \operatorname {HeunB}\left (-c +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 {\left (b c -2 \operatorname {C0} \right ) \sqrt {2}}{\sqrt {a}}, \frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )}\right ] \\ \end{align*}