Link to actual problem [9480] \[ \boxed {x^{2} y^{\prime \prime }-\left (a \,x^{2}+2\right ) y=0} \]
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 &= -\frac {{\mathrm e}^{\sqrt {a}\, x} \left (-x a +\sqrt {a}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{-\sqrt {a}\, x} y}{x a -\sqrt {a}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\sqrt {a}\, x} \left (x a +\sqrt {a}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{\sqrt {a}\, x} y}{x a +\sqrt {a}}\right ] \\ \end{align*}