Link to actual problem [7500] \[ \boxed {\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y=0} \]
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 &= \frac {26}{3}+x^{2}-6 x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\frac {26}{3}+x^{2}-6 x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{3} x^{2}-2 x +\frac {10}{3}, \underline {\hspace {1.25 ex}}\eta &= y x\right ] \\ \left [R &= \frac {y \,{\mathrm e}^{-9 \arctan \left (x -3\right )}}{\left (x^{2}-6 x +10\right )^{\frac {3}{2}}}, S \left (R \right ) &= 3 \arctan \left (x -3\right )\right ] \\ \end{align*}