Link to actual problem [12052] \[ \boxed {\tan \left (t \right ) x^{\prime \prime }-3 x^{\prime }+\left (\tan \left (t \right )+3 \cot \left (t \right )\right ) x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= \sin \left (t \right ) \end {align*}
type detected by program
{"reduction_of_order"}
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 {x}{\sin \left (t \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= t, S \left (R \right ) &= \frac {x}{\sin \left (2 t \right )}\right ] \\ \end{align*}