Link to actual problem [10995] \[ \boxed {x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y=0} \]
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 &= \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, -\frac {1}{2}+\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}\right ], \left [\frac {b \sqrt {a^{2}}+a b -2 c}{2 \sqrt {a^{2}}}\right ], \frac {1}{2}+\frac {a +2 x}{2 \sqrt {a^{2}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, -\frac {1}{2}+\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}\right ], \left [\frac {b \sqrt {a^{2}}+a b -2 c}{2 \sqrt {a^{2}}}\right ], \frac {\sqrt {a^{2}}+a +2 x}{2 \sqrt {a^{2}}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\sqrt {a^{2}}+a +2 x \right )^{1-\frac {b \sqrt {a^{2}}\, a +a^{2} b -2 c \sqrt {a^{2}}}{2 a^{2}}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {b^{2}-2 b -4 d +1}\, \sqrt {a^{2}}+a b -2 c}{2 \sqrt {a^{2}}}, \frac {1}{2}+\frac {\sqrt {b^{2}-2 b -4 d +1}\, \sqrt {a^{2}}-a b +2 c}{2 \sqrt {a^{2}}}\right ], \left [2-\frac {b \sqrt {a^{2}}+a b -2 c}{2 \sqrt {a^{2}}}\right ], \frac {1}{2}+\frac {a +2 x}{2 \sqrt {a^{2}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\sqrt {a^{2}}+a +2 x \right )^{\frac {\left (a b -2 c \right ) \sqrt {a^{2}}+a^{2} b}{2 a^{2}}} y}{\left (\sqrt {a^{2}}+a +2 x \right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-2 b -4 d +1}\, \sqrt {a^{2}}-\sqrt {a^{2}}+a b -2 c}{2 \sqrt {a^{2}}}, \frac {\sqrt {a^{2}}+\sqrt {b^{2}-2 b -4 d +1}\, \sqrt {a^{2}}-a b +2 c}{2 \sqrt {a^{2}}}\right ], \left [-\frac {\left (b -4\right ) \sqrt {a^{2}}+a b -2 c}{2 \sqrt {a^{2}}}\right ], \frac {\sqrt {a^{2}}+a +2 x}{2 \sqrt {a^{2}}}\right )}\right ] \\ \end{align*}