What are natural frequency and damped frequency of a vibrating system?

Obtain the differential equation of linear simple harmonic motion.

Two equation of two simple harmonic motion are $x=a\sin (\omega t-\alpha ) and y=b\cos (\omega t-\alpha ).$The phase difference between the two is:-

A particle is subjected to two mutually perpendicular simple harmonic motions such that its x and y coordinates are given by $x=2\sin \omega t; y=2\sin (\omega t+\frac{\mathrm{\pi }}{4})$. The path of the particle will be