Link to actual problem [5195] \[ \boxed {y^{\prime \prime }-60 y^{\prime }-900 y=5 \,{\mathrm e}^{10 x}} \]
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}^{30 \left (1+\sqrt {2}\right ) x}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-30 \left (1+\sqrt {2}\right ) x} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-30 \left (\sqrt {2}-1\right ) x}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{30 \left (\sqrt {2}-1\right ) x} y\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}