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**Unformatted text preview: **+ π
3 y = 10e−x/5 sin x + π
3 , y = ±10e−x/5 √
2. Proceeding as in the ﬁrst example, we factor out (t + 3) 2 from each term in the function
√
√
√
x(t) = (t + 3) 2 cos(2t) + (t + 3) 2 sin(2t) to get x(t) = (t + 3) 2(cos(2t) + sin(2t)). We ﬁnd
15 Take a good Diﬀerential Equations class to see this! 756 Applications of Trigonometry
√
(cos(2t) + sin(2t)) = 2 sin 2t + π , so x(t) = 2(t + 3) sin 2t + π . Graphing this on the
4
4
calculator as y = 2(x + 3) sin 2x + π , we ﬁnd the sinusoid’s amplitude growing. Since our
4
amplitude function here is A(x) = 2(x + 3) = 2x + 6, which continues to grow without bound
as x → ∞, this is hardly surprising. The phenomenon illustrated here is ‘forced’ motion.
That is, we imagine that the entire apparatus on which the spring is attached is oscillating
as well. In this case, we are witnessing a ‘resonance’ eﬀect – the frequency of the external
oscillation matches the frequency of the motion of the object on the spr...

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