Link to actual problem [10860] \[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y=0} \]
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 (2 \sqrt {a^{2}}\, a^{2} x^{2}+2 a^{3} x^{2}+3 \sqrt {a^{2}}\, a b x +3 a^{2} b x -12 \alpha \sqrt {a^{2}}\right )}{12 a^{2}}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a^{2}-a \beta +b \alpha \right ) \sqrt {a^{2}}}{a^{3}}, -\frac {9^{\frac {2}{3}} \left (b^{2}+8 \alpha \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 (2 \sqrt {a^{2}}\, a^{2} x^{2}+2 a^{3} x^{2}+3 \sqrt {a^{2}}\, a b x +3 a^{2} b x -12 \alpha \sqrt {a^{2}}\right )}{12 a^{2}}} y}{\operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a^{2}-a \beta +b \alpha \right ) \sqrt {a^{2}}}{a^{3}}, -\frac {9^{\frac {2}{3}} \left (b^{2}+8 \alpha \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 \sqrt {a^{2}}\, a^{2} x^{2}-2 a^{3} x^{2}+3 \sqrt {a^{2}}\, a b x -3 a^{2} b x -12 \alpha \sqrt {a^{2}}\right )}{12 a^{2}}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{\frac {1}{3}}}, \frac {3 \left (a^{2}-a \beta +b \alpha \right ) \sqrt {a^{2}}}{a^{3}}, -\frac {9^{\frac {2}{3}} \left (b^{2}+8 \alpha \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 {\sqrt {a^{2}}\, x \left (2 a \,x^{2} \sqrt {a^{2}}-2 a^{2} x^{2}+3 b x \sqrt {a^{2}}-3 a b x +12 \alpha \right )}{12 a^{2}}} y}{\operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{\frac {1}{3}}}, \frac {3 \left (a^{2}-a \beta +b \alpha \right ) \sqrt {a^{2}}}{a^{3}}, -\frac {9^{\frac {2}{3}} \left (b^{2}+8 \alpha \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*}