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I need to determine if $x(t) = 9\cos(2t) + 4\sin(\pi t)$ is periodic. If it is periodic I need to find the period. this what I have done \begin{align*} T_0 &= 2\pi/w\\ T_1 &= 2\pi/2 = \pi \\ T_2 &= 2\pi/\pi = 2\\ T &= T_1/T_2\\ T &= \pi/2 \end{align*} I am a little confused here, can someone check and tell me if I have the right answer, thanks

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  • $\begingroup$ Use Latex. It's unable to understand your approach. $\endgroup$
    – ABC
    May 2, 2013 at 15:59
  • $\begingroup$ why? can you please tell me what is that you do not understand, thanks $\endgroup$
    – carlos
    May 2, 2013 at 16:44
  • $\begingroup$ Those $T_i's$ were not clear now, it's clear $\endgroup$
    – ABC
    May 2, 2013 at 18:05

2 Answers 2

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Your $x(t)$ is not periodic. If we assume that $T$ is a period of $x(t)$, then it is also a period of $x''(t)=-36\cos(2t)-4\pi^2\sin(\pi t)$ and also of $4x(t)+x''(t)=(16-4\pi^2)\sin(\pi t)$ and of $\pi^2x(t)+x''(t)=(9\pi^2 -36)\cos(2t)$. Since $\pi^2$ is irrational, the coefficients $16-4\pi^2$ and $9\pi^2-36$ are nonzero, and this would imply that $T$ is both a multiple of $2$ (the periods of $\sin(\pi t)$) and of $\pi$ (the periods of $\cos(2t)$). Again, since $\pi$ is irrational, there is no common multiple of $\pi$ and $2$ (apart from $0$).

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  • $\begingroup$ I am more confuse!!! $\endgroup$
    – carlos
    May 2, 2013 at 16:03
  • $\begingroup$ Why? I thought I'd remove all possible confusion ... $\endgroup$ May 2, 2013 at 16:04
  • $\begingroup$ I guess this is too much for me to comprehend!!! $\endgroup$
    – carlos
    May 2, 2013 at 16:13
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$\cos(2t)$ is periodic with period $\pi$; i.e. it repeats after $\pi, 2\pi, 3\pi$, etc. $\sin(\pi t)$ is is periodic with period $2$; i.e. it repeats after $2,4,6$, etc.

The period of the combined function will be the least common integer multiple of $\pi$ and $2$, the smallest number appearing in both lists above. Unfortunately, there is no such common multiple.

Contrast this with the case of $\cos(2t)$ and $\sin(3t)$. The latter repeats after $2\pi/3, 4\pi/3, 6\pi/3,\ldots$. Note that $2\pi$ appears on both lists, so is the minimum period of all nontrivial linear combinations of these two functions.

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  • $\begingroup$ I am sorry but I having trouble understanding where do you get sin(3t) you meant sin(pi*t)? when you said "the period of the combined function"? what do you meant combined function exactly? the addition or multiplication of both function? $\endgroup$
    – carlos
    May 2, 2013 at 16:39
  • $\begingroup$ wait!! i think i got it, since my first period is 2, now to find the 2 period i have to replace omega (w) for the next period which is 3. right, using this formula x(y) = (t+c) so this how you get sin(3t) $\endgroup$
    – carlos
    May 2, 2013 at 16:51
  • $\begingroup$ The third paragraph of my answer is to solve a completely different problem than yours; I am showing how this problem does have a solution while yours does not. $\endgroup$
    – vadim123
    May 2, 2013 at 18:11

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