Link to actual problem [9756] \[ \boxed {y^{\prime \prime }+\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {y}{\sin \left (x \right )^{2}}=0} \]
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 \sqrt {-\csc \left (x \right )^{2}}\, \ln \left (\csc \left (x \right )-\cot \left (x \right )\right ) \sin \left (x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (\csc \left (x \right )-\cot \left (x \right )\right )^{-i \sqrt {-\csc \left (x \right )^{2}}\, \sin \left (x \right )} y\right ] \\ \end{align*}