Link to actual problem [11499] \[ \boxed {t^{2} x^{\prime \prime }+x^{\prime } t +\left (t^{2}-\frac {1}{4}\right ) x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= \frac {\cos \left (t \right )}{\sqrt {t}} \end {align*}
type detected by program
{"reduction_of_order", "second_order_bessel_ode", "second_order_change_of_variable_on_y_method_1"}
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 (t \right )}{\sqrt {t}}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {\sqrt {t}\, x}{\sin \left (t \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\cos \left (t \right )}{\sqrt {t}}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {\sqrt {t}\, x}{\cos \left (t \right )}\right ] \\ \end{align*}