Link to actual problem [7581] \[ \boxed {3 x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-\left (1+3 x \right ) y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {y}{x \left (x^{2}+20 x +70\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {x}{3}} \operatorname {hypergeom}\left (\left [3\right ], \left [-\frac {1}{3}\right ], \frac {x}{3}\right )}{x^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{3}} x^{\frac {1}{3}} y}{\operatorname {hypergeom}\left (\left [3\right ], \left [-\frac {1}{3}\right ], \frac {x}{3}\right )}\right ] \\ \end{align*}