Link to actual problem [10902] \[ \boxed {x y^{\prime \prime }+\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (-2+b \right ) x^{3} y=0} \]
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 &= {\mathrm e}^{-\frac {a \left (b -2\right ) x^{2}}{2}} \operatorname {KummerM}\left (-1+\frac {b}{2}, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a \left (b -2\right ) x^{2}}{2}} y}{\operatorname {KummerM}\left (-1+\frac {b}{2}, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {a \left (b -2\right ) x^{2}}{2}} \operatorname {KummerU}\left (-1+\frac {b}{2}, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a \left (b -2\right ) x^{2}}{2}} y}{\operatorname {KummerU}\left (-1+\frac {b}{2}, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right )}\right ] \\ \end{align*}