Link to actual problem [7148] \[ \boxed {y^{\prime \prime }-a x y^{\prime }-b x y=c \,x^{2}} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {b x}{a}} \operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {b x}{a}} y}{\operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {b x}{a}} \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {b x}{a}} y}{\operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (x \,a^{2}+2 b \right )^{2}}{2 a^{3}}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (c \left (-b x +a \right )-b^{2} y\right )}{b^{2}}\right ] \\ \end{align*}