Link to actual problem [7622] \[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (3-x \right ) y^{\prime }+4 y=0} \]
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} \left (-1+x \right )}{\left (1+x \right )^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (1+x \right )^{3} y}{x^{2} \left (-1+x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{2} \left (x \ln \left (x \right )-\ln \left (x \right )-4\right )}{\left (1+x \right )^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (1+x \right )^{3} y}{x^{2} \left (-4+\left (-1+x \right ) \ln \left (x \right )\right )}\right ] \\ \end{align*}