Calculus of a Single Variable, Chapter 6, 6.1, Section 6.1, Problem 15

y^((4)) means the fourth derivative of y. The first is -6sin(2x), the second is -12cos(2x), the third is 24sin(2x) and the fourth is 48cos(2x).
We use the known derivatives of sin(z) and cos(z) and the chain rule for z = 2x.
From the other side, 16y = 48 cos(2x), the same function, so yes, this is a solution.