Link to actual problem [11042] \[ \boxed {x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y=0} \]
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 &= x^{\frac {a -c +\sqrt {a^{2}-2 a c -4 a d +c^{2}}}{2 a}} \left (x^{2}+a \right )^{-\frac {b}{2}+1+\frac {c}{2 a}} \operatorname {hypergeom}\left (\left [\frac {3}{4}+\frac {c +\sqrt {a^{2}-2 a c -4 a d +c^{2}}}{4 a}, \frac {5}{4}+\frac {-2 a b +\sqrt {a^{2}-2 a c -4 a d +c^{2}}+c}{4 a}\right ], \left [1+\frac {\sqrt {a^{2}-2 a c -4 a d +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {a -c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}} \left (x^{2}+a \right )^{\frac {b}{2}} \left (x^{2}+a \right )^{-\frac {c}{2 a}} y}{\left (x^{2}+a \right ) \operatorname {hypergeom}\left (\left [\frac {3 a +c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{4 a}, \frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}+\left (-2 b +5\right ) a +c}{4 a}\right ], \left [\frac {2 a +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {-a +c +\sqrt {a^{2}-2 a c -4 a d +c^{2}}}{2 a}} \left (x^{2}+a \right )^{-\frac {b}{2}+1+\frac {c}{2 a}} \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {-c +\sqrt {a^{2}-2 a c -4 a d +c^{2}}}{4 a}, \frac {5}{4}-\frac {2 a b +\sqrt {a^{2}-2 a c -4 a d +c^{2}}-c}{4 a}\right ], \left [1-\frac {\sqrt {a^{2}-2 a c -4 a d +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {-a +c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}} \left (x^{2}+a \right )^{\frac {b}{2}} \left (x^{2}+a \right )^{-\frac {c}{2 a}} y}{\left (x^{2}+a \right ) \operatorname {hypergeom}\left (\left [-\frac {-3 a -c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{4 a}, \frac {-\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}+\left (-2 b +5\right ) a +c}{4 a}\right ], \left [-\frac {-2 a +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right )}\right ] \\ \end{align*}