Link to actual problem [8167] \[ \boxed {4 x^{2} y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }+\left (4 x -1\right ) y=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_y_method_1", "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 &= \frac {1}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x}\, y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-2 x}}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x}\, {\mathrm e}^{2 x} y\right ] \\ \end{align*}