Link to actual problem [13527] \[ \boxed {y^{\prime \prime }+x^{2} y^{\prime }-4 y=0} \]
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^{3}}{3}} \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{3}}{3}} y}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{3}}{3}} \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )^{2}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{3}}{3}} y}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )^{2}}d x \right )}\right ] \\ \end{align*}