Link to actual problem [10959] \[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y=0} \]
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 &= \sqrt {x}\, {\mathrm e}^{-c x} \operatorname {HeunD}\left (-4 \sqrt {b \left (-2 c +a \right )}, -1-4 \sqrt {b \left (-2 c +a \right )}, 8 \sqrt {b \left (-2 c +a \right )}, -4 \sqrt {b \left (-2 c +a \right )}+1, \frac {\sqrt {b \left (-2 c +a \right )}\, x -b}{\sqrt {b \left (-2 c +a \right )}\, x +b}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{c x} y}{\sqrt {x}\, \operatorname {HeunD}\left (-4 \sqrt {b \left (-2 c +a \right )}, -1-4 \sqrt {b \left (-2 c +a \right )}, 8 \sqrt {b \left (-2 c +a \right )}, -4 \sqrt {b \left (-2 c +a \right )}+1, \frac {\sqrt {b \left (-2 c +a \right )}\, x -b}{\sqrt {b \left (-2 c +a \right )}\, x +b}\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 {x^{2} a -c \,x^{2}-b}{x}} \operatorname {HeunD}\left (4 \sqrt {b \left (-2 c +a \right )}, -1-4 \sqrt {b \left (-2 c +a \right )}, 8 \sqrt {b \left (-2 c +a \right )}, -4 \sqrt {b \left (-2 c +a \right )}+1, \frac {\sqrt {b \left (-2 c +a \right )}\, x -b}{\sqrt {b \left (-2 c +a \right )}\, x +b}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (a -c \right ) x^{2}-b}{x}} y}{\sqrt {x}\, \operatorname {HeunD}\left (4 \sqrt {b \left (-2 c +a \right )}, -1-4 \sqrt {b \left (-2 c +a \right )}, 8 \sqrt {b \left (-2 c +a \right )}, -4 \sqrt {b \left (-2 c +a \right )}+1, \frac {\sqrt {b \left (-2 c +a \right )}\, x -b}{\sqrt {b \left (-2 c +a \right )}\, x +b}\right )}\right ] \\ \end{align*}