Link to actual problem [7768] \[ \boxed {y^{\prime \prime }+n^{2} y-\frac {6 y}{x^{2}}=0} \]
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 {\sin \left (n x \right ) x^{2} n^{2}+3 \cos \left (n x \right ) n x -3 \sin \left (n x \right )}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{\sin \left (n x \right ) x^{2} n^{2}+3 \cos \left (n x \right ) n x -3 \sin \left (n x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {-\cos \left (n x \right ) x^{2} n^{2}+3 \sin \left (n x \right ) n x +3 \cos \left (n x \right )}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{\cos \left (n x \right ) x^{2} n^{2}-3 \sin \left (n x \right ) n x -3 \cos \left (n x \right )}\right ] \\ \end{align*}