Link to actual problem [9794] \[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime } x +2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 y a x=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_3rd_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^{2} \operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{2} \operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{2} \operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) \operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) \operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )}\right ] \\ \end{align*}