Link to actual problem [15359] \[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=9 x^{2}-12 x +2} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 3] \end {align*}
type detected by program
{"kovacic", "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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{3}, \underline {\hspace {1.25 ex}}\eta &= -x^{3}+x y +\frac {2}{3} y\right ] \\ \left [R &= -\frac {\left (x^{2}-y\right ) {\mathrm e}^{-3 x}}{x^{2}}, S \left (R \right ) &= 3 \ln \left (x \right )\right ] \\ \end{align*}