Link to actual problem [11017] \[ \boxed {x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y=0} \]
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_{2} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}} \operatorname {hypergeom}\left (\left [\frac {a_{2} b_{1} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}-\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}, \frac {a_{2} b_{1} -a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}+\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}\right ], \left [1+\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {a_{2} -a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{2 a_{2}}} y}{\operatorname {hypergeom}\left (\left [\frac {a_{2} b_{1} -a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}-\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}, \frac {a_{2} b_{1} -a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}\right ], \left [\frac {a_{2} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {-a_{2} +a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{2 a_{2}}} \operatorname {hypergeom}\left (\left [-\frac {-\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -a_{2} b_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}+a_{1}}{2 a_{2}}, -\frac {\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -a_{2} b_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}+a_{1}}{2 a_{2}}\right ], \left [1-\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}-2 a_{1} a_{2} +a_{2}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {-a_{2} +a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{2 a_{2}}} y}{\operatorname {hypergeom}\left (\left [-\frac {-\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -a_{2} b_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+a_{1}}{2 a_{2}}, -\frac {\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -a_{2} b_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+a_{1}}{2 a_{2}}\right ], \left [\frac {a_{2} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right )}\right ] \\ \end{align*}