Link to actual problem [2176] \[ \boxed {y^{\prime \prime }+4 y=x^{2}} \]
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 {x}{2}\right ] \\ \left [R &= y-\frac {x^{2}}{4}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{2}-\frac {1}{2}-4 y\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (2 x^{2}-8 y-1\right )}{4}\right ] \\ \end{align*}