Link to actual problem [7550] \[ \boxed {y^{\prime \prime }+\left (x -3\right ) y^{\prime }+3 y=0} \]
type detected by program
{"kovacic"}
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 {KummerM}\left (\frac {3}{2}, \frac {1}{2}, -\frac {1}{2} x^{2}+3 x -\frac {9}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {KummerM}\left (\frac {3}{2}, \frac {1}{2}, -\frac {\left (x -3\right )^{2}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {KummerU}\left (\frac {3}{2}, \frac {1}{2}, -\frac {1}{2} x^{2}+3 x -\frac {9}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {KummerU}\left (\frac {3}{2}, \frac {1}{2}, -\frac {\left (x -3\right )^{2}}{2}\right )}\right ] \\ \end{align*}