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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.

enter image description here

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Have you heard of Eulers formula?

It says that

$$e^{ix}=\cos(x)+i\sin(x)$$

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

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