Would anyone tell me please, where is the $c_1\cos(\omega t) +c_2\sin(\omega t)$ coming from? Please see the yellow highlighted part of the image.
Have you heard of Eulers formula?
It says that
Now assuming you are familiar with the usual form involving exponential, it is simply this formula applied to that.
For more explanation, see http://mathforum.org/mathimages/index.php/Simple_Harmonic_Motion