Link to actual problem [9571] \[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+4 y^{\prime } x +\left (x^{2}+1\right ) y=0} \]
type detected by program
{"kovacic", "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 {\sin \left (x \right )}{x^{2}-1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right ) y}{\sin \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\cos \left (x \right )}{x^{2}-1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right ) y}{\cos \left (x \right )}\right ] \\ \end{align*}