Link to actual problem [8179] \[ \boxed {x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (-\frac {1}{4} x -x^{2}\right ) y^{\prime }-\frac {5 y x}{16}=0} \]
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 &= \operatorname {hypergeom}\left (\left [-\frac {5}{8}, -\frac {1}{8}\right ], \left [-\frac {1}{4}\right ], 4 x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {5}{8}, -\frac {1}{8}\right ], \left [-\frac {1}{4}\right ], 4 x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {5}{4}} \operatorname {hypergeom}\left (\left [\frac {5}{8}, \frac {9}{8}\right ], \left [\frac {9}{4}\right ], 4 x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{\frac {5}{4}} \operatorname {hypergeom}\left (\left [\frac {5}{8}, \frac {9}{8}\right ], \left [\frac {9}{4}\right ], 4 x \right )}\right ] \\ \end{align*}