Link to actual problem [14627] \[ \boxed {t^{2} y^{\prime \prime }-4 y^{\prime } t +\left (t^{2}+6\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= t^{2} \cos \left (t \right ) \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 [R &= t, S \left (R \right ) &= \frac {y}{\sin \left (t \right ) t^{2}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= t, S \left (R \right ) &= \frac {y}{t^{2} \cos \left (t \right )}\right ] \\ \end{align*}