Link to actual problem [10985] \[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+y^{\prime } a x +y c=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2"}
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 {i \sqrt {\frac {c}{x^{2} a +b}}\, \sqrt {x^{2} a +b}\, \ln \left (\sqrt {a}\, x +\sqrt {x^{2} a +b}\right )}{\sqrt {a}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (\sqrt {a}\, x +\sqrt {x^{2} a +b}\right )^{-\frac {i \sqrt {\frac {c}{x^{2} a +b}}\, \sqrt {x^{2} a +b}}{\sqrt {a}}} y\right ] \\ \end{align*}