Link to actual problem [15494] \[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=\pi ^{2}-x^{2}} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {x}{2}-\frac {1}{2}\right ] \\ \left [R &= y+\frac {x^{2}}{4}+\frac {x}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {1}{8}+\frac {x^{2}}{4}+y -\frac {\pi ^{2}}{4}\right ] \\ \left [R &= -\frac {\left (2 \pi ^{2}-2 x^{2}-4 x -8 y-3\right ) {\mathrm e}^{-x}}{8}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {\left (2 \pi ^{2}-2 x^{2}-4 x -8 y-3\right ) \left (-2 \pi ^{2}+2 x -3\right )^{-2 \pi ^{2}-1} {\mathrm e}^{-2 x}}{8}, S \left (R \right ) &= \frac {\ln \left (-2 \pi ^{2}+2 x -3\right )}{2}\right ] \\ \end{align*}