Link to actual problem [5696] \[ \boxed {y^{\prime \prime }+10 y^{\prime }+24 y=144 t^{2}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {19}{12}}, y^{\prime }\left (0\right ) = -5\right ] \end {align*}
type detected by program
{"second_order_laplace", "second_order_linear_constant_coeff"}
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 &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {72 t^{2}}{5}-\frac {6}{5}-\frac {12 y}{5}\right ] \\ \left [R &= -\frac {\left (72 t^{2}-60 t -12 y+19\right ) {\mathrm e}^{\frac {12 t}{5}}}{12}, S \left (R \right ) &= t\right ] \\ \end{align*}