Question

Here are three plots of solutions of the forced exponential diffeq y'[t] + 2.13 y[t] = 3 Sin[2 t] + Sin[4 t] with starter values on y[0] equal to -6, 0, and 7:​

G.4.b.i) Merging onto a certain curve Here are three plats of solutions of the forced exponential diffeq with 0,and 7 r 2.13 fit 13 Sin [2 t]Sin [4 t]; endtine 9; Clearly, yl, y2, y3, t, s]; starterl 7; starter3 = -6: 1[t v2[t starter2 E(-r t) y3[t] t) IntegratelE^(r s) flsl. (s. 0, tl: E*(-r t) Integrate[E^ (r s) f[sl, fs, 0, tl starter3 E(-r t) + E^(-r t) Integrate [ E^ (r s) fs], {s, 0, t}l; starterl E^(-r th E^ solutionplots = Plot [fy1t], y2t], y3t]}, (t, 0, endtime, Plotstyle -> {{Thickness [0.018], Blue, Thickness [0.014], Red, {Thickness [0.01], Orange, PlotRange ->All, AspectRatio -> 1/GoldenRatio, AxesLabel - "t", The formulas for the nlotted curves are: Chop [Expand [yl [t] ]] 0.19477136276653245*Cos [4.*t] 0.7485152690086567*Sin [2 . +t] -0.702831238505781 Cos [2.t 0.10371575067317854*5in [4.st]+ 7.897682601272314+Power [E, -2.13*t -0.702831 Cos (2. t-0.194771 Cos 4. t + 748515 Sin 2. t+0.103716 8in [4, El+7.8976 e-2.13 Chop[Expand[y2[t] ]] y2[t Chop[Expand[y3[t]1 0.7485152690086567+Sin [2.*t] 0.10371575067317854+Sin [4 . *t ] - 5.102397398727686+Power [ E, -2.13*t] 0.702831238505781*Cos [2.t] - .19477136276653245+Cos [4.t .702831 Cos [2. t] - 0.194771 Cos [4. t 0.748515 Sin [2. t0.103716 Sin [4. tl 5.1024 e2.13 All tiese sului evua d iplt s the salutiqn ples

Answer 1

Note that as

too, ekt 0 for k> 0

So removing such exponential decay terms, we get the eventual solution as

(rounded, note the absence of the decay term)

0.703 cos(2t) 0.195 sin(4t)0.749 sin (2t) 0.104 sin(4t

This is the eventual solution

$\blacksquare$

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