Link to actual problem [8050] \[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y=0} \]
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 &= 16+\frac {x^{4}+36 x^{2}+16 x +1}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} y}{x^{4}+16 x^{3}+36 x^{2}+16 x +1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (48+3 x +\frac {108}{x}+\frac {48}{x^{2}}+\frac {3}{x^{3}}\right ) \ln \left (x \right )+120+\frac {450}{x}+\frac {280}{x^{2}}+\frac {25}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} y}{25+3 \left (x^{4}+16 x^{3}+36 x^{2}+16 x +1\right ) \ln \left (x \right )+120 x^{3}+450 x^{2}+280 x}\right ] \\ \end{align*}