The differential equation of all 'Simple Harmonic Motions 

of given period $\frac{2\pi }{n}$ is

The displacement x for all $SHM$ is given by

$\begin{array}{l}x=a\cos (nt+b)\\ \Rightarrow \frac{dx}{dt}=-na\sin (nt+b)\\ \Rightarrow \frac{d^{2}x}{d{t}^2}=-n^{2}a\cos (nt+b)\\ \Rightarrow \frac{d^{2}x}{d{t}^2}=-n^{2}x\Rightarrow \frac{d^{2}x}{d{t}^2}+n^{2}x=0\end{array}$