Link to actual problem [8087] \[ \boxed {4 x^{2} \left (2 x +1\right ) y^{\prime \prime }-2 x \left (-x +4\right ) y^{\prime }-\left (7+5 x \right ) y=0} \]
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 &= \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 {5 x^{3}-10 x^{2}-40 x -16}{\left (2 x +1\right )^{\frac {5}{4}} \sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x +1\right )^{\frac {5}{4}} \sqrt {x}\, y}{5 x^{3}-10 x^{2}-40 x -16}\right ] \\ \end{align*}