You have a mass suspended on a spring. We want to know where the mass will be at any instant of time.
The physical solution
Now before we get on to the math, let us first visualize the motion by attaching a spray paint bottle as the mass.
Oh, wait that seems like a function that we are familiar with – The sinusoid.
Without even having to write down a single equation, we have found out the solution to our problem. The motion that is traced by the mass is a sinusoid.
But what do I mean by a sinusoid ?
If you took the plotted paper and tried to create that function with the help of sum of polynomials i.e x, x2, x3 … Now you this what it would like :
By taking an infinite of these polynomial sums you get the function Since this series of polynomial occurs a lot, its given the name – sine.
A note on the cosine
Since cosine is merely the sine function pushed to the side, the analogy works the same way.
Therefore we can conclude that the motion described a mass attached to a spring is a sine or a cosine function:
Hopefully this gives you an intuition on why the solution to a simple harmonic oscillator is given by the sine or the cosine function. Following runs through how you would go about getting it from solving a differential equation.
9b-Discussion notes 