Link to actual problem [651] \[ \boxed {y^{\prime \prime }+t y^{\prime }+{\mathrm e}^{-t^{2}} y=0} \]
type detected by program
{"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], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
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 {i \sqrt {{\mathrm e}^{-t^{2}}}\, {\mathrm e}^{\frac {t^{2}}{2}} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, t}{2}\right )}{2}}\right ] \\ \left [R &= t, S \left (R \right ) &= {\mathrm e}^{-\frac {i {\mathrm e}^{\frac {t^{2}}{2}} \operatorname {erf}\left (\frac {\sqrt {2}\, t}{2}\right ) \sqrt {2}\, \sqrt {\pi \,{\mathrm e}^{-t^{2}}}}{2}} y\right ] \\ \end{align*}