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00009Oooool00ol0003oooooooooo`3Hooooo`006oooool01Ol0003ooooooooooooooooo000002Ko oooo00?o0000ooooooooool0eoooool001_ooooo0_l00002ooooo`03o`000?oooooooooo02Gooooo 00?o0000ooooooooool0e_ooool001_ooooo00Go0000ooooooooooooooooo`00000Xooooo`03o`00 0?oooooooooo0=Gooooo000Kooooo`05o`000?ooooooooooooooool00000:Oooool00ol0003ooooo o`000002o`0002oooooo1Ol0000_ooooo`?o0000"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.539323, -1.80777, \ 0.0197364, 0.0228747}}] }, Open ]], Cell["\<\ Problem 1) Try to solve the problem analytically with f(t)=cos(\ \[Omega]t). Produce typical plots.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(cossol[x0_, v0_] := \ DSolve[{\(x'\)[t] \[Equal] v[t], \ \(v'\)[t]\ + 2*\[Beta]*v[t] + \ \(\[Omega]\_0\^2\) x[t] \[Equal] \ Cos[\[Omega]\ t], x[0] == x0, v[0] == v0}, {x[t], v[t]}, t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(cossol[x0, v0]\)], "Input"], Cell[BoxData[ \({{x[ t] \[Rule] \((\[ExponentialE]\^\(\(-t\)\ \((\(-\[Beta]\) - \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\) - t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \((2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\^2 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\^2 - 8\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\^4\ \[Omega]\^2 + 8\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^4\ \[Omega]\^2 - 8\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\^5\ \[Omega]\^2 + 8\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^5\ \[Omega]\^2 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\^4 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\^4 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\^3\ \[Omega]\^4 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^3\ \[Omega]\^4 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\^2\ Cos[t\ \[Omega]] - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\^2\ Cos[t\ \[Omega]] - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\^2\ Cos[ t\ \[Omega]] + 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) \ - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\^2\ Cos[ t\ \[Omega]] + \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^3\ Sin[ t\ \[Omega]] - \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\ \) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^3\ Sin[ t\ \[Omega]] - \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ \ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ \[Omega]\^3\ Sin[ t\ \[Omega]] + \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ \[Omega]\^3\ Sin[t\ \[Omega]] + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Beta]\^3\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Beta]\ \[Omega]\^2\ \[Omega]\_0\%2 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ \[Omega]\_0\%2 + 12\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%2 - 12\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%2 + 12\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^3\ \[Omega]\^2\ \[Omega]\_0\%2 - 12\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^3\ \[Omega]\^2\ \[Omega]\_0\%2 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Omega]\^4\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Omega]\^4\ \[Omega]\_0\%2 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\ \[Omega]\^4\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\ \[Omega]\^4\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2 + 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) \ - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2 - \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 + \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 + \[ExponentialE]\^\(2\ \ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 - \[ExponentialE]\^\(2\ \ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 - \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 + \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 + \[ExponentialE]\^\(2\ \ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 - \[ExponentialE]\^\(2\ \ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Beta]\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\_0\%4 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\_0\%4 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\^3\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^3\ \[Omega]\_0\%4 - 4\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Omega]\^2\ \[Omega]\_0\%4 + 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Omega]\^2\ \[Omega]\_0\%4 - 4\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\ \[Omega]\^2\ \[Omega]\_0\%4 + 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\ \[Omega]\^2\ \[Omega]\_0\%4 + \ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\ \)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4 - \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4 - \[ExponentialE]\^\(2\ \ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4 + \[ExponentialE]\^\(2\ \ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Omega]\_0\%6 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Omega]\_0\%6 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\ \[Omega]\_0\%6 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\ \[Omega]\_0\%6 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Beta]\^2\ \[Omega]\^2\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 8\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\^4\ \[Omega]\^2\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 8\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^4\ \[Omega]\^2\ \@\(\[Beta]\^2 - \[Omega]\ \_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\^4\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\^4\ \@\(\[Beta]\^2 - \[Omega]\ \_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - 4\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) \ - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 8\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\ Sin[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - \ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\ \)\)\ \[Beta]\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - \ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\ \)\)\ \[Beta]\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + \ \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\ \)\)\ \[Beta]\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + \ \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\ \))\)\)\ \[Beta]\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Beta]\^2\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Omega]\^2\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \[Omega]\ \_0\%2\) - 12\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%2\ \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\) - 12\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%2\ \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Omega]\^4\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Omega]\^4\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\ \(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 6\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\ \(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) \ + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\ \))\)\)\ x0\ \[Beta]\^2\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 4\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Omega]\^2\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Omega]\^2\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\ \(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Omega]\_0\%6\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Omega]\_0\%6\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\ \))\)/\((4\ \((\[Beta]\^2 - \[Omega]\_0\%2)\)\^\(3/2\)\ \((\(-\[Beta]\) - \ \[ImaginaryI]\ \[Omega] + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\ \((\[Beta] - \ \[ImaginaryI]\ \[Omega] + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\ \((\(-\[Beta]\ \) + \[ImaginaryI]\ \[Omega] + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\ \((\ \[Beta] + \[ImaginaryI]\ \[Omega] + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\))\), v[t] \[Rule] \((\[ExponentialE]\^\(\(-t\)\ \((\(-\[Beta]\) - \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\) - t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \((\(-4\)\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^4\ \[Omega]\^2 + 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^4\ \[Omega]\^2 + 8\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\^5\ \[Omega]\^2 - 8\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^5\ \[Omega]\^2 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\^3\ \[Omega]\^4 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^3\ \[Omega]\^4 + 6\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%2 - 6\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%2 - 12\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^3\ \[Omega]\^2\ \[Omega]\_0\%2 + 12\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^3\ \[Omega]\^2\ \[Omega]\_0\%2 + 8\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\^4\ \[Omega]\^2\ \[Omega]\_0\%2 - 8\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^4\ \[Omega]\^2\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\ \[Omega]\^4\ \[Omega]\_0\%2 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\ \[Omega]\^4\ \[Omega]\_0\%2 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\^4\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\^4\ \[Omega]\_0\%2 - \ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\ \)\)\ \[Beta]\^2\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2 + \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2 + \[ExponentialE]\^\(2\ \ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2 - \[ExponentialE]\^\(2\ \ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ \[Omega]\^2\ Cos[t\ \[Omega]]\ \[Omega]\_0\%2 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 + 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) \ - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Beta]\^2\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\^3\ \[Omega]\_0\%4 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^3\ \[Omega]\_0\%4 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Omega]\^2\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ \[Omega]\_0\%4 + 4\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\ \[Omega]\^2\ \[Omega]\_0\%4 - 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\ \[Omega]\^2\ \[Omega]\_0\%4 - 12\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%4 + 12\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%4 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Omega]\^4\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Omega]\^4\ \[Omega]\_0\%4 + \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \ Cos[t\ \[Omega]]\ \[Omega]\_0\%4 - \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4 - \[ExponentialE]\^\(2\ \ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4 + \[ExponentialE]\^\(2\ \ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4 + \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4 - \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4 - \[ExponentialE]\^\(2\ \ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4 + \[ExponentialE]\^\(2\ \ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4 - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) \ - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Omega]\_0\%6 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\_0\%6 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\ \[Omega]\_0\%6 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\ \[Omega]\_0\%6 + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\_0\%6 - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Beta]\^2\ \[Omega]\_0\%6 + 4\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Omega]\^2\ \[Omega]\_0\%6 - 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Omega]\^2\ \[Omega]\_0\%6 - \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%6 + \[ExponentialE]\^\(2\ \ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%6 + \[ExponentialE]\^\(2\ \ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%6 - \[ExponentialE]\^\(2\ \ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%6 - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ x0\ \[Omega]\_0\%8 + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ x0\ \[Omega]\_0\%8 - 4\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Beta]\^3\ \[Omega]\^2\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\^2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 8\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\^4\ \[Omega]\^2\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 8\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^4\ \[Omega]\^2\ \@\(\[Beta]\^2 - \[Omega]\ \_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\^4\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\^4\ \@\(\[Beta]\^2 - \[Omega]\ \_0\%2\) + 8\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^3\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 4\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 4\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ \[Beta]\ \[Omega]\^2\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 12\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%2\ \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\) - 12\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\^2\ \[Omega]\_0\%2\ \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Omega]\^4\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Omega]\^4\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 6\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\ \(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ \[Omega]\^2\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) \ - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\^2\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\ \(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\^3\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%2\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Beta]\^2\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) \ + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\%2\ \))\)\)\ v0\ \[Beta]\^2\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) + 4\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Omega]\^2\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 4\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Omega]\^2\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Beta]\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Beta]\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Beta]\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\ \(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Beta]\ Cos[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \[Beta] + 3\ t\ \((\(-\[Beta]\) - \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + 2\ \[ExponentialE]\^\(2\ t\ \[Beta] + 2\ t\ \((\(-\[Beta]\ \) - \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) + \[ExponentialE]\^\(2\ t\ \[Beta] + t\ \((\(-\[Beta]\) - \@\ \(\[Beta]\^2 - \[Omega]\_0\%2\))\) + 3\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ \[Omega]\ Sin[ t\ \[Omega]]\ \[Omega]\_0\%4\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(2\ t\ \((\(-\[Beta]\) - \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\) + t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \[Omega]\_0\ \%2\))\)\)\ v0\ \[Omega]\_0\%6\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\) - 2\ \[ExponentialE]\^\(t\ \((\(-\[Beta]\) - \@\(\[Beta]\^2 \ - \[Omega]\_0\%2\))\) + 2\ t\ \((\(-\[Beta]\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\))\)\)\ v0\ \[Omega]\_0\%6\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\ \))\)/\((4\ \((\[Beta]\^2 - \[Omega]\_0\%2)\)\^\(3/2\)\ \((\(-\[Beta]\) - \ \[ImaginaryI]\ \[Omega] + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\ \((\[Beta] - \ \[ImaginaryI]\ \[Omega] + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\ \((\(-\[Beta]\ \) + \[ImaginaryI]\ \[Omega] + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\ \((\ \[Beta] + \[ImaginaryI]\ \[Omega] + \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\))\)}}\ \)], "Output"] }, Open ]], Cell["\<\ This does not look useful, but Mathematica doesn't care. 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ooooooooo`0booooo`;o000000Coooooo`000?oooooo00002Oooool00ol0003oooooooooo`0:oooo o`?o000000Koooooo`000?l0003o0000ooooool00002ooooo`03o`000?oooooooooo00;o00001?oo ool00ol0003oooooooooo`02ooooo`;o00001oooool7o`0000Sooooo00Co0000oooooooooooooooo 0_l000000ooooooo0000ooooo`0"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.11819, -7.64114, \ 0.0405086, 0.624232}}] }, Open ]], Cell["\<\ If you examine the analytic expression, you will see that there are \ both positive and negative exponents that become large when \[Beta] is large. \ This leads to the subtraction of large numbers, and unless you force \ Mathematica to use more than machine precision there will be large roundoff \ errors. Let's try simplifying the expression.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cossol[x0_, v0_] = \ FullSimplify[ DSolve[{\(x'\)[t] \[Equal] v[t], \ \(v'\)[t]\ + 2*\[Beta]*v[t] + \ \(\[Omega]\_0\^2\) x[t] \[Equal] \ Cos[\[Omega]\ t], x[0] == x0, v[0] == v0}, {x[t], v[t]}, t]]\)], "Input"], Cell[BoxData[ \({{x[ t] \[Rule] \((\[ExponentialE]\^\(\(-t\)\ \((\[Beta] + \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\))\)\)\ \((\((\(-1\) + \[ExponentialE]\^\(2\ t\ \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\)\))\)\ \((\[Beta]\ \((\(-1\) + 4\ \[Beta]\ \((v0 + x0\ \[Beta])\))\)\ \[Omega]\^2 + \((v0 \ + x0\ \[Beta])\)\ \[Omega]\^4 - \((\[Beta] + 2\ \((v0 + x0\ \[Beta])\)\ \[Omega]\^2)\)\ \ \[Omega]\_0\%2 + \((v0 + x0\ \[Beta])\)\ \[Omega]\_0\%4)\) + \@\(\ \[Beta]\^2 - \[Omega]\_0\%2\)\ \((\((1 + \[ExponentialE]\^\(2\ t\ \@\(\[Beta]\ \^2 - \[Omega]\_0\%2\)\))\)\ \((\[Omega]\^2\ \((1 + x0\ \((4\ \[Beta]\^2 + \ \[Omega]\^2)\))\) - \((1 + 2\ x0\ \[Omega]\^2)\)\ \[Omega]\_0\%2 + x0\ \[Omega]\_0\%4)\) + 2\ \[ExponentialE]\^\(t\ \((\[Beta] + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \((2\ \[Beta]\ \[Omega]\ Sin[ t\ \[Omega]] + Cos[t\ \[Omega]]\ \((\(-\[Omega]\^2\) + \ \[Omega]\_0\%2)\))\))\))\))\)/\((2\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\)\ \((4\ \ \[Beta]\^2\ \[Omega]\^2 + \[Omega]\^4 - 2\ \[Omega]\^2\ \[Omega]\_0\%2 + \[Omega]\_0\%4)\))\), v[t] \[Rule] \((\[ExponentialE]\^\(\(-t\)\ \((\[Beta] + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \((\(-\((\(-1\) + \ \[ExponentialE]\^\(2\ t\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\)\))\)\)\ \((\[Beta]\ \ \[Omega]\^2\ \((\(-2\)\ \[Beta] + 4\ v0\ \[Beta]\^2 + v0\ \[Omega]\^2)\) + \[Omega]\^2\ \((1 - 2\ v0\ \[Beta] + 4\ x0\ \[Beta]\^2 + x0\ \[Omega]\^2)\)\ \[Omega]\_0\%2 + \ \((\(-1\) + v0\ \[Beta] - 2\ x0\ \[Omega]\^2)\)\ \[Omega]\_0\%4 + x0\ \[Omega]\_0\%6)\) + \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\)\ \((\((1 + \[ExponentialE]\^\(2\ t\ \@\(\[Beta]\^2 - \ \[Omega]\_0\%2\)\))\)\ \((2\ \[Beta]\ \((\(-1\) + 2\ v0\ \[Beta])\)\ \[Omega]\^2 + v0\ \[Omega]\^4 - 2\ v0\ \[Omega]\^2\ \[Omega]\_0\%2 + v0\ \[Omega]\_0\%4)\) + 2\ \[ExponentialE]\^\(t\ \((\[Beta] + \ \@\(\[Beta]\^2 - \[Omega]\_0\%2\))\)\)\ \[Omega]\ \((2\ \[Beta]\ \[Omega]\ \ Cos[t\ \[Omega]] + Sin[t\ \[Omega]]\ \((\[Omega]\^2 - \ \[Omega]\_0\%2)\))\))\))\))\)/\((2\ \@\(\[Beta]\^2 - \[Omega]\_0\%2\)\ \((4\ \ \[Beta]\^2\ \[Omega]\^2 + \[Omega]\^4 - 2\ \[Omega]\^2\ \[Omega]\_0\%2 + \ \[Omega]\_0\%4)\))\)}}\)], "Output"] }, Open ]], Cell["\<\ Note the = sign rather than :=. 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