Link to actual problem [6375] \[ \boxed {y^{\prime \prime }-2 y^{\prime }-5 y=x} \]
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}{5}}\right ] \\ \left [R &= y+\frac {x}{5}, 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}-\frac {5 y}{2}\right ] \\ \left [R &= \frac {\left (25 y+5 x -2\right ) {\mathrm e}^{\frac {5 x}{2}}}{25}, S \left (R \right ) &= x\right ] \\ \end{align*}