Link to actual problem [7588] \[ \boxed {x^{2} \left (2+x \right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (-8 x +2\right ) y=0} \]
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 &= 60+\frac {40 x^{4}-160 x^{3}+8 x +1}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{40 x^{4}-160 x^{3}+60 x^{2}+8 x +1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (-x -2\right )^{\frac {3}{4}} \left (105 x^{\frac {3}{2}} \left (40 x^{4}-160 x^{3}+60 x^{2}+8 x +1\right ) \operatorname {arcsinh}\left (\frac {\sqrt {x}\, \sqrt {2}}{2}\right )+\frac {\sqrt {2+x}\, x^{2} \left (8 x^{5}+328 x^{4}-13974 x^{3}+26734 x^{2}-805 x -105\right )}{2}\right )}{\left (2+x \right )^{\frac {3}{4}} x^{\frac {7}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2+x \right )^{\frac {3}{4}} x^{\frac {7}{2}} y}{4 \left (-x -2\right )^{\frac {3}{4}} \left (\left (\frac {105 x^{\frac {3}{2}}}{4}+210 x^{\frac {5}{2}}+1575 x^{\frac {7}{2}}-4200 x^{\frac {9}{2}}+1050 x^{\frac {11}{2}}\right ) \operatorname {arcsinh}\left (\frac {\sqrt {x}\, \sqrt {2}}{2}\right )+\sqrt {2+x}\, x^{2} \left (x^{5}+41 x^{4}-\frac {6987}{4} x^{3}+\frac {13367}{4} x^{2}-\frac {805}{8} x -\frac {105}{8}\right )\right )}\right ] \\ \end{align*}