Link to actual problem [9759] \[ \boxed {y^{\prime \prime }+\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\frac {\left (-17 \sin \left (x \right )^{2}-1\right ) y}{4 \sin \left (x \right )^{2}}=0} \]
type detected by program
{"second_order_change_of_variable_on_y_method_1"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
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 &= \frac {\sinh \left (2 x \right )}{\sqrt {\sin \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {\sin \left (x \right )}\, y}{\sinh \left (2 x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\cosh \left (2 x \right )}{\sqrt {\sin \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {\sin \left (x \right )}\, y}{\cosh \left (2 x \right )}\right ] \\ \end{align*}