Link to actual problem [8102] \[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y=0} \]
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 {\left (x^{2}+1\right )^{\frac {5}{2}} \left (x^{2}+8\right )}{x^{7}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{7} y}{\left (x^{2}+1\right )^{\frac {5}{2}} \left (x^{2}+8\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {35 x^{6}+140 x^{4}+168 x^{2}+64}{x^{7}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{7} y}{35 x^{6}+140 x^{4}+168 x^{2}+64}\right ] \\ \end{align*}