Equation of a simple harmonic motion(SHM)

[johncena]

According to my textbook, the equation of a simple harmonic motion(SHM) is given by,
y = a sin$$\omega$$t
or, x = a cos$$\omega$$t
But how did these equations came?
Like this there are somany equations in my textbook without derivations.And studying these byheart makes the subject very boring.

 

[nnnm4]

A simple harmonic oscillator feels a restoring force that is proportional to it's displacement, so Newton's second law tells us:

$$m\frac{d^2x}{dt^2} = -kx$$

or

$$\frac{d^2x}{dt^2} = -\omega_o^2x$$

where

$$\omega_o = \sqrt{\frac{k}{m}}$$

Now, there are some techniques for solving this differential equation, but we can just think about it intuitively. What function, if you take it's derivative twice, gives back the negative of the same function times a constant. Well cosine and sine both posses those properties. So those are both solutions, and any linear combination of them will be a solution.