Link to actual problem [10836] \[ \boxed {y^{\prime \prime }+y^{\prime } a -\left (b \,x^{2}+c \right ) y=0} \]
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 \operatorname {KummerM}\left (\frac {3}{4}+\frac {a^{2}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) {\mathrm e}^{-\frac {x \left (x \sqrt {b}+a \right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (x \sqrt {b}+a \right )}{2}} y}{x \operatorname {KummerM}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {KummerU}\left (\frac {3}{4}+\frac {a^{2}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) {\mathrm e}^{-\frac {x \left (x \sqrt {b}+a \right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (x \sqrt {b}+a \right )}{2}} y}{x \operatorname {KummerU}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right )}\right ] \\ \end{align*}