Link to actual problem [10862] \[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+x \left (a b \,x^{2}+b c +2 a \right ) y=0} \]
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}^{-\frac {x \left (2 a \,x^{2} \sqrt {a^{2}}+2 a^{2} x^{2}-3 b x \sqrt {a^{2}}+3 a b x +6 c \sqrt {a^{2}}+6 a c \right )}{12 a}} \operatorname {HeunT}\left (0, \frac {3 \sqrt {a^{2}}}{a}, \frac {9^{\frac {2}{3}} \left (4 a c -b^{2}\right )}{12 \left (a^{4}\right )^{\frac {1}{3}}}, \frac {3^{\frac {2}{3}} a \left (2 x a -b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (\left (x^{2} a -\frac {3}{2} b x +3 c \right ) \sqrt {a^{2}}+a \left (x^{2} a +\frac {3}{2} b x +3 c \right )\right )}{6 a}} y}{\operatorname {HeunT}\left (0, \frac {3 \sqrt {a^{2}}}{a}, \frac {9^{\frac {2}{3}} \left (4 a c -b^{2}\right )}{12 \left (a^{4}\right )^{\frac {1}{3}}}, \frac {3^{\frac {2}{3}} a \left (2 x a -b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x \left (2 a \,x^{2} \sqrt {a^{2}}-2 a^{2} x^{2}-3 b x \sqrt {a^{2}}-3 a b x +6 c \sqrt {a^{2}}-6 a c \right )}{12 a}} \operatorname {HeunT}\left (0, -\frac {3 \sqrt {a^{2}}}{a}, \frac {9^{\frac {2}{3}} \left (4 a c -b^{2}\right )}{12 \left (a^{4}\right )^{\frac {1}{3}}}, -\frac {3^{\frac {2}{3}} a \left (2 x a -b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (\left (-x^{2} a +\frac {3}{2} b x -3 c \right ) \sqrt {a^{2}}+a \left (x^{2} a +\frac {3}{2} b x +3 c \right )\right )}{6 a}} y}{\operatorname {HeunT}\left (0, -\frac {3 \sqrt {a^{2}}}{a}, \frac {9^{\frac {2}{3}} \left (4 a c -b^{2}\right )}{12 \left (a^{4}\right )^{\frac {1}{3}}}, -\frac {3^{\frac {2}{3}} \left (x a -\frac {b}{2}\right ) a}{3 \left (a^{2}\right )^{\frac {5}{6}}}\right )}\right ] \\ \end{align*}