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I have the equation for simple harmonic motion ##\displaystyle \frac{d^2x}{dx^2} + k^2 x = 0##. I have a simple question. Do we need to make an assumption about the sign of ##k## before we solve this? We have that the roots satisfy ##r^2=-k^2##. So ##r=\pm i \sqrt{k^2}##. Do I need to assume ##k## is either positive or negative before I can proceed?