Link to actual problem [13707] \[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=18 x^{2}+3 x +4} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{3}-\frac {8 x^{2}}{3}+\frac {4 y}{3}\right ] \\ \left [R &= -\left (2 x^{2}+3 x -y+2\right ) {\mathrm e}^{-\frac {4 x}{3}}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{3}-\frac {3}{8}, \underline {\hspace {1.25 ex}}\eta &= -2 x^{3}+\frac {13}{24}+x y -\frac {5}{6} y\right ] \\ \left [R &= \frac {\left (-2 x^{2}-3 x +y-2\right ) {\mathrm e}^{-3 x}}{\left (8 x -9\right )^{\frac {7}{8}}}, S \left (R \right ) &= 3 \ln \left (8 x -9\right )\right ] \\ \end{align*}