Link to actual problem [2758] \[ \boxed {y^{\prime \prime }-4 y^{\prime }+6 y=7 \,{\mathrm e}^{2 x}} \]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{2 x} \sin \left (\sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-2 x} y}{\sin \left (\sqrt {2}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{2 x} \cos \left (\sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-2 x} y}{\cos \left (\sqrt {2}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}