Link to actual problem [8103] \[ \boxed {4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y=0} \]
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 &= \frac {3 x^{2}+2}{x^{\frac {5}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {5}{2}} y}{3 x^{2}+2}\right ] \\ \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 {3}{2}}}{x^{\frac {5}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {5}{2}} y}{\left (x^{2}+1\right )^{\frac {3}{2}}}\right ] \\ \end{align*}