Link to actual problem [7575] \[ \boxed {x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y=0} \]
type detected by program
{"kovacic"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
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}^{-\ln \left (10 x^{2}+x +5\right )-\frac {2 \sqrt {199}\, \arctan \left (\frac {\left (20 x +1\right ) \sqrt {199}}{199}\right )}{995}+\frac {\ln \left (x \right )}{5}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (10 x^{2}+x +5\right ) {\mathrm e}^{\frac {2 \sqrt {199}\, \arctan \left (\frac {\left (20 x +1\right ) \sqrt {199}}{199}\right )}{995}} y}{x^{\frac {1}{5}}}\right ] \\ \end{align*}