Link to actual problem [10898] \[ \boxed {x y^{\prime \prime }-\left (a x +1\right ) y^{\prime }-b \,x^{2} \left (b x +a \right ) y=0} \]
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 &= {\mathrm e}^{-\frac {b \,x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {b \,x^{2}}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \pi \,\operatorname {erf}\left (\frac {-2 b x -a}{2 \sqrt {-b}}\right ) a \,{\mathrm e}^{-\frac {2 b^{2} x^{2}+a^{2}}{4 b}}-2 \sqrt {\pi }\, \sqrt {-b}\, {\mathrm e}^{\frac {x \left (b x +2 a \right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {2 b^{2} x^{2}+a^{2}}{4 b}} y}{-2 \sqrt {\pi }\, \sqrt {-b}\, {\mathrm e}^{\frac {1}{2} b \,x^{2}+x a} {\mathrm e}^{\frac {2 b^{2} x^{2}+a^{2}}{4 b}}-a \,\operatorname {erf}\left (\frac {2 b x +a}{2 \sqrt {-b}}\right ) \pi }\right ] \\ \end{align*}