Link to actual problem [7901] \[ \boxed {t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{3} y=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2"}
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 &= {\mathrm e}^{-\frac {t^{2}}{4}} \cos \left (\frac {t^{2} \sqrt {3}}{4}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {t^{2}}{4}} y}{\cos \left (\frac {t^{2} \sqrt {3}}{4}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {t^{2}}{4}} \sin \left (\frac {t^{2} \sqrt {3}}{4}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {t^{2}}{4}} y}{\sin \left (\frac {t^{2} \sqrt {3}}{4}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{t}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \frac {t^{2}}{2}\right ] \\ \end{align*}