Link to actual problem [10854] \[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y=0} \]
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 &= \operatorname {hypergeom}\left (\left [\frac {1}{4}+\frac {a^{3}-2 a^{2} \gamma +2 a b \beta -2 \alpha \,b^{2}-4 a \alpha +8 \alpha \gamma -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (x \,a^{2}+a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {\left (\left (x a +2 b \right ) \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+\left (a^{2}-4 \alpha \right ) \left (x \,a^{2}+2 a b -4 \alpha x -4 \beta \right )\right ) x}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (\left (x a +2 b \right ) \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+\left (a^{2}-4 \alpha \right ) \left (x \,a^{2}+2 a b -4 \alpha x -4 \beta \right )\right ) x}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}} y}{\operatorname {hypergeom}\left (\left [\frac {\left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \gamma +\left (2 b \beta -4 \alpha \right ) a +\left (-2 b^{2}+8 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (x \,a^{2}+a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {\left (\left (x a +2 b \right ) \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+\left (a^{2}-4 \alpha \right ) \left (x \,a^{2}+2 a b -4 \alpha x -4 \beta \right )\right ) x}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}} \operatorname {hypergeom}\left (\left [\frac {3}{4}+\frac {a^{3}-2 a^{2} \gamma +2 a b \beta -2 \alpha \,b^{2}-4 a \alpha +8 \alpha \gamma -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (x \,a^{2}+a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ) \left (x \,a^{2}+a b -4 \alpha x -2 \beta \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (\left (x a +2 b \right ) \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+\left (a^{2}-4 \alpha \right ) \left (x \,a^{2}+2 a b -4 \alpha x -4 \beta \right )\right ) x}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}} y}{\operatorname {hypergeom}\left (\left [\frac {3 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \gamma +\left (2 b \beta -4 \alpha \right ) a +\left (-2 b^{2}+8 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (x \,a^{2}+a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ) \left (x \,a^{2}+a b -4 \alpha x -2 \beta \right )}\right ] \\ \end{align*}