Link to actual problem [14046] \[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }+5 y^{\prime }+y={\mathrm e}^{x}} \]
type detected by program
{"higher_order_linear_constant_coefficients_ODE"}
type detected by Maple
[[_3rd_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}^{-\frac {\left (\left (404+60 \sqrt {69}\right )^{{2}/{3}}-4 \left (404+60 \sqrt {69}\right )^{{1}/{3}}-44\right ) x}{6 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {\left (\left (404+60 \sqrt {69}\right )^{{2}/{3}}-4 \left (404+60 \sqrt {69}\right )^{{1}/{3}}-44\right ) \left (-101+15 \sqrt {69}\right ) \left (404+60 \sqrt {69}\right )^{{2}/{3}} x}{127776}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {\left (-44+\left (404+60 \sqrt {69}\right )^{{2}/{3}}+8 \left (404+60 \sqrt {69}\right )^{{1}/{3}}\right ) x}{12 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (404+60 \sqrt {69}\right )^{{2}/{3}}+44\right ) x}{12 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {\left (-101+15 \sqrt {69}\right ) \left (404+60 \sqrt {69}\right )^{{2}/{3}} \left (-44+\left (404+60 \sqrt {69}\right )^{{2}/{3}}+8 \left (404+60 \sqrt {69}\right )^{{1}/{3}}\right ) x}{255552}} y}{\sin \left (\frac {\sqrt {3}\, \left (\left (404+60 \sqrt {69}\right )^{{2}/{3}}+44\right ) x \left (-101+15 \sqrt {69}\right ) \left (404+60 \sqrt {69}\right )^{{2}/{3}}}{255552}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {\left (-44+\left (404+60 \sqrt {69}\right )^{{2}/{3}}+8 \left (404+60 \sqrt {69}\right )^{{1}/{3}}\right ) x}{12 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (404+60 \sqrt {69}\right )^{{2}/{3}}+44\right ) x}{12 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {\left (-101+15 \sqrt {69}\right ) \left (404+60 \sqrt {69}\right )^{{2}/{3}} \left (-44+\left (404+60 \sqrt {69}\right )^{{2}/{3}}+8 \left (404+60 \sqrt {69}\right )^{{1}/{3}}\right ) x}{255552}} y}{\cos \left (\frac {\sqrt {3}\, \left (\left (404+60 \sqrt {69}\right )^{{2}/{3}}+44\right ) x \left (-101+15 \sqrt {69}\right ) \left (404+60 \sqrt {69}\right )^{{2}/{3}}}{255552}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (-5 y+{\mathrm e}^{x}\right )}{5}\right ] \\ \end{align*}