Link to actual problem [10843] \[ \boxed {y^{\prime \prime }+y^{\prime } a x +y b=0} \]
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 {x^{2} a}{2}} \operatorname {KummerM}\left (1-\frac {b}{2 a}, \frac {3}{2}, \frac {x^{2} a}{2}\right ) x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2} a}{2}} y}{\operatorname {KummerM}\left (\frac {2 a -b}{2 a}, \frac {3}{2}, \frac {x^{2} a}{2}\right ) x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{2} a}{2}} \operatorname {KummerU}\left (1-\frac {b}{2 a}, \frac {3}{2}, \frac {x^{2} a}{2}\right ) x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2} a}{2}} y}{\operatorname {KummerU}\left (\frac {2 a -b}{2 a}, \frac {3}{2}, \frac {x^{2} a}{2}\right ) x}\right ] \\ \end{align*}