Link to actual problem [9739] \[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x \ln \left (x \right )}-\ln \left (x \right )^{2} y=0} \]
type detected by program
{"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}^{i \left (\sqrt {-\ln \left (x \right )^{2}}\, x -\frac {\sqrt {-\ln \left (x \right )^{2}}\, x}{\ln \left (x \right )}\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {i \sqrt {-\ln \left (x \right )^{2}}\, x \left (-1+\ln \left (x \right )\right )}{\ln \left (x \right )}} y\right ] \\ \end{align*}