1. **State the problem:** Solve the ordinary differential equation $$y'' + 16y = 0$$ where $y''$ denotes the second derivative of $y$ with respect to $x$. 2. **Identify the type of equation:** This is a second-order linear homogeneous differential equation with constant coefficients. 3. **Write the characteristic equation:** Replace $y''$ by $r^2$ and $y$ by 1 to get $$r^2 + 16 = 0$$ 4. **Solve the characteristic equation:** $$r^2 = -16$$ $$r = \pm 4i$$ where $i$ is the imaginary unit. 5. **Write the general solution:** For complex roots $\alpha \pm \beta i$, the general solution is $$y = C_1 \cos(\beta x) + C_2 \sin(\beta x)$$ Here, $\alpha = 0$ and $\beta = 4$, so $$y = C_1 \cos(4x) + C_2 \sin(4x)$$ 6. **Interpretation:** The solution represents oscillatory behavior with angular frequency 4. **Final answer:** $$y = C_1 \cos(4x) + C_2 \sin(4x)$$