Link to actual problem [12318] \[ \boxed {4 y^{\prime \prime }-4 y^{\prime }+y=t^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = -12, y^{\prime }\left (0\right ) = 7] \end {align*}
type detected by program
{"second_order_laplace", "second_order_linear_constant_coeff", "linear_second_order_ode_solved_by_an_integrating_factor"}
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 {t^{2}}{4}+\frac {y}{4}+2\right ] \\ \left [R &= -\left (t^{2}+8 t -y+24\right ) {\mathrm e}^{-\frac {t}{4}}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {\left (t^{2}+8 t -y+24\right ) {\mathrm e}^{-\frac {t}{2}}}{t -6}, S \left (R \right ) &= \frac {\ln \left (t -6\right )}{2}\right ] \\ \end{align*}