Link to actual problem [5139] \[ \boxed {y^{\prime \prime }+25 y=5 x^{2}+x} \]
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 {1}{25}+\frac {2 x}{5}\right ] \\ \left [R &= -\frac {x^{2}}{5}-\frac {x}{25}+y, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {1}{5} x^{2}-\frac {1}{25} x +\frac {2}{125}+y\right ] \\ \\ \end{align*}