Link to actual problem [15287] \[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=x^{2}} \]
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\right ] \\ \left [R &= -\frac {\left (2 x^{2}+4 x -8 y+3\right ) {\mathrm e}^{-x}}{8}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x -\frac {3}{2}, \underline {\hspace {1.25 ex}}\eta &= -\frac {1}{2} x^{3}+2 x y -2 y\right ] \\ \left [R &= -\frac {\left (2 x^{2}+4 x -8 y+3\right ) {\mathrm e}^{-2 x}}{8 \left (2 x -3\right )}, S \left (R \right ) &= \ln \left (2 x -3\right )\right ] \\ \end{align*}