Link to actual problem [10901] \[ \boxed {x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+y b \,x^{3}=0} \]
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 &= x^{2} \operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{\frac {1}{4}}}, i \left (-b \right )^{\frac {1}{4}} x \right ) {\mathrm e}^{\frac {x \left (x \sqrt {-b}+2 a \right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x a -\frac {x^{2} \sqrt {-b}}{2}} y}{x^{2} \operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{\frac {1}{4}}}, i \left (-b \right )^{\frac {1}{4}} x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{2} \operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{\frac {1}{4}}}, i \left (-b \right )^{\frac {1}{4}} x \right ) {\mathrm e}^{\frac {x \left (x \sqrt {-b}+2 a \right )}{2}} \left (\int \frac {{\mathrm e}^{-x^{2} \sqrt {-b}}}{\operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{\frac {1}{4}}}, i \left (-b \right )^{\frac {1}{4}} x \right )^{2} x^{3}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x a -\frac {x^{2} \sqrt {-b}}{2}} y}{x^{2} \operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{\frac {1}{4}}}, i \left (-b \right )^{\frac {1}{4}} x \right ) \left (\int \frac {{\mathrm e}^{-x^{2} \sqrt {-b}}}{\operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{\frac {1}{4}}}, i \left (-b \right )^{\frac {1}{4}} x \right )^{2} x^{3}}d x \right )}\right ] \\ \end{align*}