Link to actual problem [12347] \[ \boxed {y^{\prime \prime \prime }-5 y^{\prime \prime }+y^{\prime }-y=-t^{2}+2 t -10} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0] \end {align*}
type detected by program
{"higher_order_laplace"}
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 (116+6 \sqrt {78}\right )^{{2}/{3}}+5 \left (116+6 \sqrt {78}\right )^{{1}/{3}}+22\right ) t}{3 \left (116+6 \sqrt {78}\right )^{{1}/{3}}}}\right ] \\ \left [R &= t, S \left (R \right ) &= {\mathrm e}^{\frac {\left (\left (116+6 \sqrt {78}\right )^{{2}/{3}}+5 \left (116+6 \sqrt {78}\right )^{{1}/{3}}+22\right ) \left (116+6 \sqrt {78}\right )^{{2}/{3}} \left (-58+3 \sqrt {78}\right ) t}{15972}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {\left (22+\left (116+6 \sqrt {78}\right )^{{2}/{3}}-10 \left (116+6 \sqrt {78}\right )^{{1}/{3}}\right ) t}{6 \left (116+6 \sqrt {78}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (116+6 \sqrt {78}\right )^{{2}/{3}}-22\right ) t}{6 \left (116+6 \sqrt {78}\right )^{{1}/{3}}}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {\left (116+6 \sqrt {78}\right )^{{2}/{3}} \left (-58+3 \sqrt {78}\right ) \left (22+\left (116+6 \sqrt {78}\right )^{{2}/{3}}-10 \left (116+6 \sqrt {78}\right )^{{1}/{3}}\right ) t}{31944}} y}{\cos \left (\frac {\sqrt {3}\, \left (\left (116+6 \sqrt {78}\right )^{{2}/{3}}-22\right ) t \left (116+6 \sqrt {78}\right )^{{2}/{3}} \left (-58+3 \sqrt {78}\right )}{31944}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {\left (22+\left (116+6 \sqrt {78}\right )^{{2}/{3}}-10 \left (116+6 \sqrt {78}\right )^{{1}/{3}}\right ) t}{6 \left (116+6 \sqrt {78}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (116+6 \sqrt {78}\right )^{{2}/{3}}-22\right ) t}{6 \left (116+6 \sqrt {78}\right )^{{1}/{3}}}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= -\frac {{\mathrm e}^{-\frac {\left (116+6 \sqrt {78}\right )^{{2}/{3}} \left (-58+3 \sqrt {78}\right ) \left (22+\left (116+6 \sqrt {78}\right )^{{2}/{3}}-10 \left (116+6 \sqrt {78}\right )^{{1}/{3}}\right ) t}{31944}} y}{\sin \left (\frac {\sqrt {3}\, \left (\left (116+6 \sqrt {78}\right )^{{2}/{3}}-22\right ) t \left (116+6 \sqrt {78}\right )^{{2}/{3}} \left (-58+3 \sqrt {78}\right )}{31944}\right )}\right ] \\ \end{align*}
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