Link to actual problem [8024] \[ \boxed {4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y=0} \]
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 {\sqrt {x}}{\left (x^{2}+1\right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{\frac {3}{2}} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {x}\, \operatorname {arcsinh}\left (x \right )}{\left (x^{2}+1\right )^{\frac {3}{2}}}-\frac {1}{\left (x^{2}+1\right ) \sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{\frac {5}{2}} \sqrt {x}\, y}{-\left (x^{2}+1\right )^{\frac {3}{2}}+\left (x^{3}+x \right ) \operatorname {arcsinh}\left (x \right )}\right ] \\ \end{align*}