Link to actual problem [7631] \[ \boxed {4 x^{2} y^{\prime \prime }+2 y^{\prime } x^{3}+\left (3 x^{2}+1\right ) y=0} \]
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 &= \sqrt {x}\, {\mathrm e}^{-\frac {x^{2}}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, {\mathrm e}^{-\frac {x^{2}}{4}} \operatorname {expIntegral}_{1}\left (-\frac {x^{2}}{4}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} y}{\sqrt {x}\, \operatorname {expIntegral}_{1}\left (-\frac {x^{2}}{4}\right )}\right ] \\ \end{align*}