Link to actual problem [9744] \[ \boxed {y^{\prime \prime }+\frac {\left (\sin \left (x \right )^{2}-\cos \left (x \right )\right ) y^{\prime }}{\sin \left (x \right )}+y \sin \left (x \right )^{2}=0} \]
type detected by program
{"second_order_change_of_variable_on_x_method_2"}
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 &= {\mathrm e}^{\frac {\cos \left (x \right )}{2}} \sin \left (\frac {\sqrt {3}\, \cos \left (x \right )}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {\cos \left (x \right )}{2}} y}{\sin \left (\frac {\sqrt {3}\, \cos \left (x \right )}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {\cos \left (x \right )}{2}} \cos \left (\frac {\sqrt {3}\, \cos \left (x \right )}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {\cos \left (x \right )}{2}} y}{\cos \left (\frac {\sqrt {3}\, \cos \left (x \right )}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{\sin \left (x \right )}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \\ \end{align*}