Link to actual problem [7959] \[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (4+x \right ) y^{\prime }+\left (2-x \right ) y=0} \]
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 &= x +6+\frac {3}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{x^{2}+6 x +3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 51+\frac {3 x^{3} \ln \left (x \right )+18 \ln \left (x \right ) x^{2}+9 x \ln \left (x \right )+48 x +1}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{1+3 \left (x^{3}+6 x^{2}+3 x \right ) \ln \left (x \right )+51 x^{2}+48 x}\right ] \\ \end{align*}