SHM with acceleration at mean position

Question

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?

Answer 1

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 $y$.

Answer 2

The mean position is $c/k$: this is just SHM about a point other than the origin, with the solution being:

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