Link to actual problem [8194] \[ \boxed {y^{\prime \prime }-a^{2} y-\frac {6 y}{x^{2}}=0} \]
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}^{x a} \left (a^{2} x^{2}-3 x a +3\right )}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} {\mathrm e}^{-x a} y}{a^{2} x^{2}-3 x a +3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-x a} \left (a^{2} x^{2}+3 x a +3\right )}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} {\mathrm e}^{x a} y}{a^{2} x^{2}+3 x a +3}\right ] \\ \end{align*}