Link to actual problem [7568] \[ \boxed {y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\right ) y=0} \]
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}^{\frac {x^{2}}{2}} \left (x^{6}-15 x^{4}+45 x^{2}-15\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{2}}{2}} y}{x^{6}-15 x^{4}+45 x^{2}-15}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\sqrt {2}\, x^{6}-15 \sqrt {2}\, x^{4}+45 \sqrt {2}\, x^{2}-15 \sqrt {2}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {x^{2}}{2}} \operatorname {erfi}\left (\frac {\sqrt {2}\, x}{2}\right )+\left (-2 x^{5}+28 x^{3}-66 x \right ) {\mathrm e}^{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {\pi }\, \left (x^{6}-15 x^{4}+45 x^{2}-15\right ) \operatorname {erfi}\left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\, {\mathrm e}^{\frac {x^{2}}{2}}-2 \,{\mathrm e}^{x^{2}} x \left (x^{2}-11\right ) \left (x^{2}-3\right )}\right ] \\ \end{align*}