Link to actual problem [10866] \[ \boxed {y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+2 a \right ) y^{\prime }+a^{2} \left (x^{3} b +1\right ) y=0} \]
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 [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 {{\mathrm e}^{-\frac {x^{3} b \left (3 x a +4\right )}{12}}}{\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 {{\mathrm e}^{-\frac {1}{4} a b \,x^{4}-\frac {1}{3} b \,x^{3}}}{\left (x a +1\right )^{2}}d x \right )}\right ] \\ \end{align*}