Link to actual problem [10857] \[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (a \,x^{2}+b -c \right ) y=0} \]
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 &= \operatorname {HeunT}\left (0, -\frac {3 \sqrt {a^{2}}}{a}, \frac {a 9^{\frac {2}{3}} \left (b -2 c \right )}{3 \left (a^{4}\right )^{\frac {1}{3}}}, \frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{-\frac {x \left (a \,x^{2} \sqrt {a^{2}}+a^{2} x^{2}+3 b \sqrt {a^{2}}+3 a b -6 a c \right )}{6 \sqrt {a^{2}}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (\left (x^{2} a +3 b \right ) \sqrt {a^{2}}+a \left (x^{2} a +3 b -6 c \right )\right )}{6 \sqrt {a^{2}}}} y}{\operatorname {HeunT}\left (0, -\frac {3 \sqrt {a^{2}}}{a}, \frac {a 9^{\frac {2}{3}} \left (b -2 c \right )}{3 \left (a^{4}\right )^{\frac {1}{3}}}, \frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunT}\left (0, \frac {3 \sqrt {a^{2}}}{a}, \frac {a 9^{\frac {2}{3}} \left (b -2 c \right )}{3 \left (a^{4}\right )^{\frac {1}{3}}}, -\frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{-\frac {x \left (a \,x^{2} \sqrt {a^{2}}-a^{2} x^{2}+3 b \sqrt {a^{2}}-3 a b +6 a c \right )}{6 \sqrt {a^{2}}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x \left (\left (-x^{2} a -3 b \right ) \sqrt {a^{2}}+a \left (x^{2} a +3 b -6 c \right )\right )}{6 \sqrt {a^{2}}}} y}{\operatorname {HeunT}\left (0, \frac {3 \sqrt {a^{2}}}{a}, \frac {a 9^{\frac {2}{3}} \left (b -2 c \right )}{3 \left (a^{4}\right )^{\frac {1}{3}}}, -\frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right )}\right ] \\ \end{align*}