Link to actual problem [8193] \[ \boxed {u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u=0} \]
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 {\cos \left (x a \right ) a x -\sin \left (x a \right )}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} u}{\cos \left (x a \right ) a x -\sin \left (x a \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\cos \left (x a \right )+\sin \left (x a \right ) a x}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} u}{\cos \left (x a \right )+\sin \left (x a \right ) a x}\right ] \\ \end{align*}