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Suppose we have an equation of motion as $$\frac{d^2x}{dt^2} = -kx + c,$$ then can it be called a SHM? Since acceleration is still proportional to displacement. But then, how will we define the mean position?

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You could rewrite your equation as $\frac{d^2x}{dt^2} = -k \left (x + \frac c k\right )$

Setting $y = x + \frac c k$ gives $\frac{d^2y}{dt^2} = -ky$

The equation of shm which can be solved for $x$ and hence $y$.

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  • $\begingroup$ I see, so it's just like shifting of origin concept? As in, 'y' is the displacement from mean position, and 'x' is the absolute position. Is that right? Thanks! $\endgroup$ – Shodai Feb 17 '16 at 16:04
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The mean position is $c/k$: this is just SHM about a point other than the origin, with the solution being:

$$ x(t) = a_1 \sin\left(\sqrt{k}t\right) + a_2 \cos\left(\sqrt{k} t\right) + c/k $$

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