Link to actual problem [8231] \[ \boxed {x \left (2+x \right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }-4 y=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
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 &= x^{2}+2 x +\frac {1}{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2}+2 x +\frac {1}{2}}\right ] \\ \end{align*}