Link to actual problem [9618] \[ \boxed {\left (27 x^{2}+4\right ) y^{\prime \prime }+27 y^{\prime } x -3 y=0} \]
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 {3}{27 x^{2}+4}}\, \sqrt {27 x^{2}+4}\, \sqrt {3}\, \operatorname {arcsinh}\left (\frac {3 \sqrt {3}\, x}{2}\right )}{9}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {i \sqrt {-\frac {1}{27 x^{2}+4}}\, \sqrt {27 x^{2}+4}\, \operatorname {arcsinh}\left (\frac {3 \sqrt {3}\, x}{2}\right )}{3}} y\right ] \\ \end{align*}