Link to actual problem [8144] \[ \boxed {y^{\prime \prime }+x y^{\prime }-4 y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {y}{x^{4}+6 x^{2}+3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right ) \left (x^{4}+6 x^{2}+3\right )+\left (\sqrt {2}\, x^{3}+5 \sqrt {2}\, x \right ) {\mathrm e}^{-\frac {x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right ) \left (x^{4}+6 x^{2}+3\right ) {\mathrm e}^{\frac {x^{2}}{2}}+\sqrt {2}\, x \left (x^{2}+5\right )}\right ] \\ \end{align*}