Link to actual problem [7594] \[ \boxed {2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y=0} \]
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 {x^{\frac {7}{2}}}{\left (2 x^{2}+4\right )^{\frac {9}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {8 \left (x^{2}+2\right )^{3} y}{x^{\frac {7}{2}} \left (2 x^{2}+4\right )^{\frac {3}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1\right ], \left [-\frac {1}{2}\right ], -\frac {x^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1\right ], \left [-\frac {1}{2}\right ], -\frac {x^{2}}{2}\right )}\right ] \\ \end{align*}