Question

solve y" + 4y = 16Cos(2x) bit stuck on calculating the particular solution

Answer 1

if the homogenous solution is part of the right-hand side, then you should
multiply your "guess" for yp by x to make it "independent"

in this case, you can write y_h as A cos(2x) + B sin(2x)
which contains your right-hand side. so you need to try
yp = C*x*sin(2x) + D*x*cos(2x)