Link to actual problem [10912] \[ \boxed {x y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+a x -1\right ) y^{\prime }+a^{2} b \,x^{3} y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x a} y}{x a +1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-x a} \left (x a +1\right ) \left (\int \frac {x \,{\mathrm e}^{-\frac {x \left (2 a b \,x^{2}+3 b x -6 a \right )}{6}}}{\left (x a +1\right )^{2}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x a} y}{\left (x a +1\right ) \left (\int \frac {x \,{\mathrm e}^{-\frac {1}{3} a b \,x^{3}-\frac {1}{2} b \,x^{2}+x a}}{\left (x a +1\right )^{2}}d x \right )}\right ] \\ \end{align*}