Link to actual problem [8328] \[ \boxed {y^{\prime \prime }-\left (\frac {x^{2}}{4}-\frac {11}{2}\right ) y=0} \]
type detected by program
{"kovacic", "second_order_bessel_ode"}
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 &= {\mathrm e}^{-\frac {x^{2}}{4}} \operatorname {hypergeom}\left (\left [-2\right ], \left [\frac {3}{2}\right ], \frac {x^{2}}{2}\right ) x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} y}{\operatorname {hypergeom}\left (\left [-2\right ], \left [\frac {3}{2}\right ], \frac {x^{2}}{2}\right ) x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{2}}{4}} \operatorname {hypergeom}\left (\left [-\frac {5}{2}\right ], \left [\frac {1}{2}\right ], \frac {x^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} y}{\operatorname {hypergeom}\left (\left [-\frac {5}{2}\right ], \left [\frac {1}{2}\right ], \frac {x^{2}}{2}\right )}\right ] \\ \end{align*}