Link to actual problem [13241] \[ \boxed {y^{\prime \prime }+16 y=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= {\frac {1}{16}}\right ] \\ \left [R &= y-\frac {t}{16}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= t, S \left (R \right ) &= -\frac {\ln \left (-16 y+t \right )}{16}\right ] \\ \end{align*}