Link to actual problem [10861] \[ \boxed {y^{\prime \prime }+\left (a b \,x^{2}+b x +2 a \right ) y^{\prime }+a^{2} \left (b \,x^{2}+1\right ) y=0} \]
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}^{\left (-\frac {a^{6} b^{6}}{6 \left (a^{2} b^{2}\right )^{\frac {5}{2}}}-\frac {a b}{6}\right ) x^{3}+\left (-\frac {a^{5} b^{6}}{4 \left (a^{2} b^{2}\right )^{\frac {5}{2}}}-\frac {b}{4}\right ) x^{2}-x a} \operatorname {HeunT}\left (\frac {b 3^{\frac {2}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}, -\frac {6 a b}{\sqrt {a^{2} b^{2}}}, -\frac {b^{2} 9^{\frac {2}{3}}}{12 \left (a^{4} b^{4}\right )^{\frac {1}{3}}}, \frac {3^{\frac {2}{3}} a \,b^{2} \left (2 x a +1\right )}{6 \left (a^{2} b^{2}\right )^{\frac {5}{6}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (a b +\sqrt {a^{2} b^{2}}\right ) x^{3}}{6}} {\mathrm e}^{\frac {\left (a b +\sqrt {a^{2} b^{2}}\right ) x^{2}}{4 a}} {\mathrm e}^{x a} y}{\operatorname {HeunT}\left (\frac {b 3^{\frac {2}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}, -\frac {6 a b}{\sqrt {a^{2} b^{2}}}, -\frac {b^{2} 9^{\frac {2}{3}}}{12 \left (a^{4} b^{4}\right )^{\frac {1}{3}}}, \frac {3^{\frac {2}{3}} a \,b^{2} \left (2 x a +1\right )}{6 \left (a^{2} b^{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}^{\left (\frac {a^{6} b^{6}}{6 \left (a^{2} b^{2}\right )^{\frac {5}{2}}}-\frac {a b}{6}\right ) x^{3}+\left (\frac {a^{5} b^{6}}{4 \left (a^{2} b^{2}\right )^{\frac {5}{2}}}-\frac {b}{4}\right ) x^{2}-x a} \operatorname {HeunT}\left (\frac {b 3^{\frac {2}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}, \frac {6 a b}{\sqrt {a^{2} b^{2}}}, -\frac {b^{2} 9^{\frac {2}{3}}}{12 \left (a^{4} b^{4}\right )^{\frac {1}{3}}}, -\frac {3^{\frac {2}{3}} a \,b^{2} \left (2 x a +1\right )}{6 \left (a^{2} b^{2}\right )^{\frac {5}{6}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {\left (-a b +\sqrt {a^{2} b^{2}}\right ) x^{3}}{6}} {\mathrm e}^{-\frac {\left (-a b +\sqrt {a^{2} b^{2}}\right ) x^{2}}{4 a}} {\mathrm e}^{x a} y}{\operatorname {HeunT}\left (\frac {b 3^{\frac {2}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}, \frac {6 a b}{\sqrt {a^{2} b^{2}}}, -\frac {b^{2} 9^{\frac {2}{3}}}{12 \left (a^{4} b^{4}\right )^{\frac {1}{3}}}, -\frac {\left (x a +\frac {1}{2}\right ) 3^{\frac {2}{3}} b^{2} a}{3 \left (a^{2} b^{2}\right )^{\frac {5}{6}}}\right )}\right ] \\ \end{align*}