Link to actual problem [9613] \[ \boxed {x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (-1+a \right ) y=0} \]
type detected by program
{"unknown"}
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 &= \operatorname {hypergeom}\left (\left [\frac {a}{2}, \frac {a}{2}-\frac {1}{2}\right ], \left [a \right ], 4 x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [\frac {a}{2}, \frac {a}{2}-\frac {1}{2}\right ], \left [a \right ], 4 x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{1-a} \operatorname {hypergeom}\left (\left [1-\frac {a}{2}, -\frac {a}{2}+\frac {1}{2}\right ], \left [-a +2\right ], 4 x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{a} y}{x \operatorname {hypergeom}\left (\left [1-\frac {a}{2}, -\frac {a}{2}+\frac {1}{2}\right ], \left [-a +2\right ], 4 x \right )}\right ] \\ \end{align*}