Link to actual problem [7533] \[ \boxed {4 y^{\prime \prime } x^{2}-4 x \left (1+x \right ) y^{\prime }+\left (3+2 x \right ) y=0} \]
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 &= \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}\, {\mathrm e}^{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} y}{\sqrt {x}}\right ] \\ \end{align*}