Link to actual problem [5970] \[ \boxed {6 y^{\prime \prime }+5 y^{\prime }-6 y=x} \]
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}{6}}\right ] \\ \left [R &= y+\frac {x}{6}, 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 {6 y}{5}+\frac {x}{5}\right ] \\ \left [R &= \frac {\left (36 y+6 x +5\right ) {\mathrm e}^{-\frac {6 x}{5}}}{36}, S \left (R \right ) &= x\right ] \\ \end{align*}