Link to actual problem [10838] \[ \boxed {y^{\prime \prime }+y^{\prime } a +b x \left (-x^{3} b +a x +2\right ) y=0} \]
type detected by program
{"kovacic"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
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^{3}}{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {b \,x^{3}}{3}} y\right ] \\ \end{align*}