Link to actual problem [9630] \[ \boxed {\left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y=0} \]
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 {1}{2}-\frac {-d +\sqrt {a^{2}-2 a d -4 a g +d^{2}}}{2 a}, -\frac {1}{2}+\frac {d +\sqrt {a^{2}-2 a d -4 a g +d^{2}}}{2 a}\right ], \left [\frac {d \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}\, a -2 a f +b d}{2 a^{2} \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}}\right ], -\frac {2 \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}\, x \,a^{2}+\sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}\, b a -4 a c +b^{2}}{2 \left (4 a c -b^{2}\right )}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {a -d +\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}}{2 a}, \frac {-a +d +\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}}{2 a}\right ], \left [\frac {d \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a -2 a f +b d}{2 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}\right ], \frac {\left (-2 x \,a^{2}-a b \right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+4 a c -b^{2}}{8 a c -2 b^{2}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\left (2 \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}\, x \,a^{2}+\sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}\, b a -4 a c +b^{2}\right )}^{1-\frac {d \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}\, a -2 a f +b d}{2 a^{2} \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {a^{2}-2 a d -4 a g +d^{2}}\, a \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}-2 a f +b d}{2 a^{2} \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}}, \frac {1}{2}+\frac {\sqrt {a^{2}-2 a d -4 a g +d^{2}}\, a \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}+2 a f -b d}{2 a^{2} \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}}\right ], \left [2-\frac {d \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}\, a -2 a f +b d}{2 a^{2} \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}}\right ], -\frac {2 \sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}\, x \,a^{2}+\sqrt {\frac {b^{2}}{a^{2}}-\frac {4 c}{a}}\, b a -4 a c +b^{2}}{2 \left (4 a c -b^{2}\right )}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\left (2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, x \,a^{2}+\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, b a -4 a c +b^{2}\right )}^{\frac {d \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a -2 a f +b d}{2 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}} y}{\left (2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, x \,a^{2}+\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, b a -4 a c +b^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {a \left (a -\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+2 a f -b d}{2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}, \frac {a \left (a +\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+2 a f -b d}{2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}\right ], \left [\frac {4 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}-d \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a +2 a f -b d}{2 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}\right ], \frac {\left (-2 x \,a^{2}-a b \right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+4 a c -b^{2}}{8 a c -2 b^{2}}\right )}\right ] \\ \end{align*}