Link to actual problem [9737] \[ \boxed {y^{\prime \prime }+\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}+\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )}=0} \]
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 &= x^{\frac {1}{2}+\frac {q}{2}+\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {hypergeom}\left (\left [-\frac {-p -q -\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [1+\frac {\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {q}{2}} x^{-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} y}{\sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {p +q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q +\sqrt {q^{2}+2 q +4 s +1}-\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [\frac {b +\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{2}+\frac {q}{2}-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {hypergeom}\left (\left [-\frac {-p -q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q -\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [1-\frac {\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {q}{2}} x^{\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} y}{\sqrt {x}\, \operatorname {hypergeom}\left (\left [-\frac {-p -q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q -\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [\frac {b -\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )}\right ] \\ \end{align*}