Link to actual problem [7552] \[ \boxed {\left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (x +1\right ) y^{\prime }+60 y=0} \]
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 &= \frac {5}{12}+\frac {x \left (2+x \right )}{6}, \underline {\hspace {1.25 ex}}\eta &= y x\right ] \\ \left [R &= \frac {y \,{\mathrm e}^{2 \sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \left (1+x \right )}{3}\right )}}{8 x^{6}+48 x^{5}+156 x^{4}+304 x^{3}+390 x^{2}+300 x +125}, S \left (R \right ) &= 2 \sqrt {6}\, \arctan \left (\frac {\left (4 x +4\right ) \sqrt {6}}{12}\right )\right ] \\ \end{align*}