Link to actual problem [7608] \[ \boxed {4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }-4 x \left (-3 x^{2}-3 x +1\right ) y^{\prime }+3 \left (x^{2}-x +1\right ) y=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_y_method_1"}
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 {\sqrt {x}}{x^{2}+3 x +1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+3 x +1\right ) y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{\frac {3}{2}}}{x^{2}+3 x +1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+3 x +1\right ) y}{x^{\frac {3}{2}}}\right ] \\ \end{align*}