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The force exerted on a mass, $m$, by an ideal spring, characterized by the spring constant $k$, is described by the equation:

$$F=-k\, x$$

Which of the following is a correct expression of Newton's second law for this force?

$m\frac{dx}{dt}+k\,x=0$

$\frac{d^2x}{dt^2}+\frac{k}{m}\,x=0$

$\frac{d^2x}{dt^2}-\frac{k}{m}\,x=0$

$\frac{d^2x}{dt^2}-\frac{m}{k}\,x=0$