Question

Find the particular solution to y ' = 4sin(x) given the general solution is y = c - 4cos(x) and the initial condition y of pi over 2 equals 2

Answer 1

The general solutions always have some additive/multiplicative constant, that you must fix in the particular solution.

In order to do so, you need to impose that the particular solution passes through a certain point. In your case, you have

$y(x) = c-4\cos(x)$

and you want

$y\left(\dfrac{\pi}{2}\right) = 2$

Put everything together, and you have

$y\left(\dfrac{\pi}{2}\right) = c-4\cos\left(\dfrac{\pi}{2}\right) = c = 2$

Since the cosine is zero in the chosen point. So, we've fixed the value of the constant, and the particular solution is found:

$y(x) = 2-4\cos(x)$