Link to actual problem [8020] \[ \boxed {2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y=0} \]
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 {1}{\left (x^{2}+1\right )^{\frac {1}{4}} x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (x^{2}+1\right )^{\frac {1}{4}} x^{\frac {3}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {hypergeom}\left (\left [1, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {hypergeom}\left (\left [1, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -x^{2}\right )}\right ] \\ \end{align*}