Link to actual problem [11041] \[ \boxed {a \,x^{2} \left (x -1\right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y=0} \]
type detected by program
{"second_order_bessel_ode"}
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 {\sqrt {a -4 d}}{2 \sqrt {a}}} \left (-1+x \right )^{\frac {1}{2}-\frac {\sqrt {a -4 b -4 c -4 d}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {a -4 b -4 c -4 d}-\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}, \frac {1}{2}+\frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [1+\frac {\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {\sqrt {a -4 d}}{2 \sqrt {a}}} \left (-1+x \right )^{\frac {\sqrt {a -4 b -4 c -4 d}}{2 \sqrt {a}}} y}{\sqrt {x}\, \sqrt {-1+x}\, \operatorname {hypergeom}\left (\left [-\frac {\sqrt {a -4 b -4 c -4 d}-\sqrt {a}-\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}, \frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}+\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [\frac {\sqrt {a}+\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (-1+x \right )^{\frac {1}{2}-\frac {\sqrt {a -4 b -4 c -4 d}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {-\sqrt {a -4 b -4 c -4 d}-\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}, \frac {1}{2}-\frac {\sqrt {a -4 b -4 c -4 d}+\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [1-\frac {\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right ) x^{\frac {1}{2}-\frac {\sqrt {a -4 d}}{2 \sqrt {a}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right )^{\frac {\sqrt {a -4 b -4 c -4 d}}{2 \sqrt {a}}} x^{\frac {\sqrt {a -4 d}}{2 \sqrt {a}}} y}{\sqrt {-1+x}\, \operatorname {hypergeom}\left (\left [\frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}-\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}, -\frac {\sqrt {a -4 b -4 c -4 d}-\sqrt {a}+\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [\frac {\sqrt {a}-\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right ) \sqrt {x}}\right ] \\ \end{align*}