Link to actual problem [8086] \[ \boxed {x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (9+13 x \right ) y^{\prime }+\left (7+5 x \right ) y=0} \]
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 &= \frac {143 x^{2}+104 x +20}{x^{7}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{7} y}{143 x^{2}+104 x +20}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (35 x^{3}-45 x^{2}+36 x -20\right ) \left (2 x +1\right )^{\frac {7}{2}}}{x^{7}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{7} y}{\left (35 x^{3}-45 x^{2}+36 x -20\right ) \left (2 x +1\right )^{\frac {7}{2}}}\right ] \\ \end{align*}