Link to actual problem [10865] \[ \boxed {y^{\prime \prime }+\left (a \,x^{3}+b x \right ) y^{\prime }+2 \left (2 a \,x^{2}+b \right ) y=0} \]
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 {HeunB}\left (-\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {\left (x^{2} a +2 b \right ) x^{2}}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (x^{2} a +2 b \right ) x^{2}}{4}} y}{\operatorname {HeunB}\left (-\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {HeunB}\left (\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {\left (x^{2} a +2 b \right ) x^{2}}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (x^{2} a +2 b \right ) x^{2}}{4}} y}{x \operatorname {HeunB}\left (\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right )}\right ] \\ \end{align*}