Link to actual problem [7586] \[ \boxed {x^{2} \left (4 x +3\right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }-\left (4 x +3\right ) y=0} \]
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 {48 x^{2}+32 x +7}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} y}{48 x^{2}+32 x +7}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [3, 5\right ], \left [\frac {13}{3}\right ], -\frac {4 x}{3}\right ) \left (3+4 x \right )^{\frac {11}{3}} x^{\frac {1}{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [3, 5\right ], \left [\frac {13}{3}\right ], -\frac {4 x}{3}\right ) \left (3+4 x \right )^{\frac {11}{3}} x^{\frac {1}{3}}}\right ] \\ \end{align*}