Link to actual problem [7039] \[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=9 x^{2}+4} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = 8] \end {align*}
type detected by program
{"kovacic", "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 {9 x^{2}}{4}+\frac {1}{2}+\frac {3 y}{4}\right ] \\ \left [R &= -\left (3 x^{2}+8 x -y+10\right ) {\mathrm e}^{-\frac {3 x}{4}}, S \left (R \right ) &= x\right ] \\ \end{align*}