Link to actual problem [224] \[ \boxed {2 y^{\prime \prime }+4 y^{\prime }+7 y=x^{2}} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {8}{49}+\frac {2 x}{7}\right ] \\ \left [R &= y-\frac {x^{2}}{7}+\frac {8 x}{49}, 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 {x^{2}}{4}-\frac {1}{7}-\frac {7 y}{4}\right ] \\ \left [R &= -\frac {\left (49 x^{2}-56 x -343 y+4\right ) {\mathrm e}^{\frac {7 x}{4}}}{343}, S \left (R \right ) &= x\right ] \\ \end{align*}