For this problem, you actually don't need to do any calculus. You are told what the solution is:
y(x) = C-2cos(x). The particular solution is found by making the initial condition work. That is, we need
y(pi/3) = 1.
1 = y(pi/3) = C-2cos(pi/3),
C = 1+2cos(pi/3) = 1+2(1/2) = 1+1 = 2.
So, y(x) = 2(1-cos(x)).