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All these solutions eventually merge onto a certain curve. 

Give the formula for this curve and add its plot to the solution plot.

286 p21.jpg

G. 4.b. i) Merging onto a certain curve
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:
In[89]:=
r = 2.13;
Clear [f, t];
f[t_] = 3 sin[2 +] + Sin[4 +];
endtime = 9;
Clear[y, y1, y2, y3, t, s];
starter1 = 7;
starter2 = 0;
starter3 = -6;
y1[t_] = starter1 =^(-r t) + E^(-r t) Integrate[E^(r s) f[s], {s, 0, t}];
y2[t_] = starter2 E^(-r t) + E^(-r t) Integrate[E^(r s) f[s], {s, 0, t}];
y3[t_] = starter3 E^(-r t) + E^(-r t) Integrate[E^(r s) f[s], {s, 0, t}];
solutionplots = Plot[{yi[t], y2[t], y3[t]), {t, 0, endtime}, PlotStyle -> {{Thickness[0.018], Blue},
{Thickness [0. 014], Red}, {Thickness[0.01], Orange}}, PlotRange -> All, AspectRatio -> 1/GoldenRatio, AxesLabel
-> {"t", ""}]
Out[100]=
2
2
8

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