Link to actual problem [8146] \[ \boxed {6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (12 x +1\right ) y=0} \]
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}^{-3 x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{3 x} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {3 \,{\mathrm e}^{-3 x} x \Gamma \left (\frac {5}{6}\right )-3 \,{\mathrm e}^{-3 x} x \Gamma \left (\frac {5}{6}, -3 x \right )-\left (-243 x^{5}\right )^{\frac {1}{6}}}{3 \sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{3 x} y}{-\frac {\left (-243 x^{5}\right )^{\frac {1}{6}} {\mathrm e}^{3 x}}{3}+x \left (\Gamma \left (\frac {5}{6}\right )-\Gamma \left (\frac {5}{6}, -3 x \right )\right )}\right ] \\ \end{align*}