Link to actual problem [7606] \[ \boxed {28 x^{2} \left (1-3 x \right ) y^{\prime \prime }-7 x \left (5+9 x \right ) y^{\prime }+7 \left (2+9 x \right ) y=0} \]
type detected by program
{"kovacic"}
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 {x^{2}}{3 x -1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (3 x -1\right ) y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{\frac {1}{4}}}{3 x -1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (3 x -1\right ) y}{x^{\frac {1}{4}}}\right ] \\ \end{align*}