Link to actual problem [9764] \[ \boxed {y^{\prime \prime }-\frac {\left (3 \sin \left (x \right )^{2}+1\right ) y^{\prime }}{\cos \left (x \right ) \sin \left (x \right )}-\frac {y \sin \left (x \right )^{2}}{\cos \left (x \right )^{2}}=0} \]
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 &= \cos \left (x \right )^{-\frac {3}{2}+\frac {\sqrt {13}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \cos \left (x \right )^{\frac {3}{2}} \cos \left (x \right )^{-\frac {\sqrt {13}}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \cos \left (x \right )^{-\frac {3}{2}-\frac {\sqrt {13}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \cos \left (x \right )^{\frac {3}{2}} \cos \left (x \right )^{\frac {\sqrt {13}}{2}} y\right ] \\ \end{align*}