1. **Problem statement:** Find the general solution to the differential equation $$y'' + 16y = 0$$. 2. **Formula and approach:** This is a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation is $$r^2 + 16 = 0$$. 3. **Solve the characteristic equation:** $$r^2 = -16$$ $$r = \pm 4i$$ 4. **Interpretation:** Since the roots are purely imaginary, the general solution is a combination of sine and cosine functions: $$y = C_1 \cos(4x) + C_2 \sin(4x)$$ 5. **Explanation:** The constants $C_1$ and $C_2$ are arbitrary and determined by initial conditions. The frequency of oscillation is 4, corresponding to the coefficient under the square root in the characteristic roots. **Final answer:** $$y = C_1 \cos(4x) + C_2 \sin(4x)$$