Link to actual problem [9658] \[ \boxed {y^{\prime \prime }-\frac {\left (x -4\right ) y^{\prime }}{2 x \left (x -2\right )}+\frac {\left (x -3\right ) y}{2 x^{2} \left (x -2\right )}=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_y_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 &= \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x^{2}-2 x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x^{2}-2 x}}\right ] \\ \end{align*}