Link to actual problem [11002] \[ \boxed {\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y=0} \]
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 {b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 a_{2} c_{0} +b_{1}^{2}}}{2 a_{2}}, -\frac {1}{2}-\frac {-b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 a_{2} c_{0} +b_{1}^{2}}}{2 a_{2}}\right ], \left [\frac {b_{1} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}\, a_{2} -2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}}\right ], \frac {\left (-2 a_{2}^{2} x -a_{2} b_{2} \right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 c_{2} a_{2} -b_{2}^{2}}{8 c_{2} a_{2} -2 b_{2}^{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}\right ], \left [\frac {b_{1} \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2} -2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}\right ], \frac {\left (-2 a_{2}^{2} x -a_{2} b_{2} \right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 c_{2} a_{2} -b_{2}^{2}}{8 c_{2} a_{2} -2 b_{2}^{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 {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}\, x \,a_{2}^{2}+\sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}\, b_{2} a_{2} -4 c_{2} a_{2} +b_{2}^{2}\right )}^{1-\frac {b_{1} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}\, a_{2} -2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 a_{2} c_{0} +b_{1}^{2}}\, a_{2} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}-2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}}, \frac {1}{2}+\frac {\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 a_{2} c_{0} +b_{1}^{2}}\, a_{2} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}+2 a_{2} c_{1} -b_{1} b_{2}}{2 a_{2}^{2} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}}\right ], \left [2-\frac {b_{1} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}\, a_{2} -2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {-\frac {4 c_{2} a_{2} -b_{2}^{2}}{a_{2}^{2}}}}\right ], \frac {\left (-2 a_{2}^{2} x -a_{2} b_{2} \right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 c_{2} a_{2} -b_{2}^{2}}{8 c_{2} a_{2} -2 b_{2}^{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\left (2 \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, x \,a_{2}^{2}+\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, b_{2} a_{2} -4 c_{2} a_{2} +b_{2}^{2}\right )}^{\frac {b_{1} \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2} -2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}} y}{\left (2 \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, x \,a_{2}^{2}+\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, b_{2} a_{2} -4 c_{2} a_{2} +b_{2}^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {\frac {a_{2} \left (a_{2} -\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}{2}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}, \frac {\frac {a_{2} \left (a_{2} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}{2}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}\right ], \left [\frac {2 a_{2} \left (a_{2} -\frac {b_{1}}{4}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}\right ], \frac {\left (-2 a_{2}^{2} x -a_{2} b_{2} \right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 c_{2} a_{2} -b_{2}^{2}}{8 c_{2} a_{2} -2 b_{2}^{2}}\right )}\right ] \\ \end{align*}