Link to actual problem [9661] \[ \boxed {y^{\prime \prime }+\frac {\left (\left (1+a \right ) x -1\right ) y^{\prime }}{x \left (x -1\right )}+\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (x -1\right )}=0} \]
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 &= \left (-1+x \right )^{1-a} x^{\frac {c}{2}} \operatorname {hypergeom}\left (\left [-\frac {a}{2}-\frac {b}{2}+\frac {c}{2}+1, -\frac {a}{2}+\frac {b}{2}+\frac {c}{2}+1\right ], \left [c +1\right ], x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right )^{a} x^{-\frac {c}{2}} y}{\left (-1+x \right ) \operatorname {hypergeom}\left (\left [-\frac {a}{2}-\frac {b}{2}+\frac {c}{2}+1, -\frac {a}{2}+\frac {b}{2}+\frac {c}{2}+1\right ], \left [c +1\right ], x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {c}{2}} \left (-1+x \right )^{1-a} \operatorname {hypergeom}\left (\left [-\frac {a}{2}-\frac {b}{2}-\frac {c}{2}+1, -\frac {a}{2}+\frac {b}{2}-\frac {c}{2}+1\right ], \left [-c +1\right ], x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {c}{2}} \left (-1+x \right )^{a} y}{\left (-1+x \right ) \operatorname {hypergeom}\left (\left [-\frac {a}{2}-\frac {b}{2}-\frac {c}{2}+1, -\frac {a}{2}+\frac {b}{2}-\frac {c}{2}+1\right ], \left [-c +1\right ], x\right )}\right ] \\ \end{align*}