Link to actual problem [10899] \[ \boxed {x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+\left (x^{3} b +a^{2} x +a \right ) y=0} \]
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 {x \left (x \sqrt {-b}+2 a \right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-x a -\frac {x^{2} \sqrt {-b}}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x \left (x \sqrt {-b}-2 a \right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {x^{2} \sqrt {-b}}{2}-x a} y\right ] \\ \end{align*}