1. **State the problem:** Solve the differential equation $y(4) + 2y + y = 0$. 2. **Clarify notation:** Here, $y(4)$ means the fourth derivative of $y$ with respect to $x$, so the equation is $$y^{(4)} + 2y + y = 0.$$ Simplify the terms: $$y^{(4)} + 3y = 0.$$ 3. **Form the characteristic equation:** Assume a solution of the form $y = e^{rx}$. Substitute into the differential equation to get the characteristic polynomial: $$r^4 + 3 = 0.$$ 4. **Solve the characteristic equation:** $$r^4 = -3.$$ Rewrite as $$r^4 = 3e^{i\pi}$$ (since $-3 = 3e^{i\pi}$). 5. **Find the roots:** The fourth roots of $3e^{i\pi}$ are given by $$r = 3^{1/4} e^{i(\frac{\pi + 2k\pi}{4})}$$ for $k=0,1,2,3$. Calculate each root: - For $k=0$: $$r_0 = 3^{1/4} e^{i\pi/4} = 3^{1/4} (\cos(\pi/4) + i\sin(\pi/4))$$ - For $k=1$: $$r_1 = 3^{1/4} e^{i3\pi/4} = 3^{1/4} (\cos(3\pi/4) + i\sin(3\pi/4))$$ - For $k=2$: $$r_2 = 3^{1/4} e^{i5\pi/4} = 3^{1/4} (\cos(5\pi/4) + i\sin(5\pi/4))$$ - For $k=3$: $$r_3 = 3^{1/4} e^{i7\pi/4} = 3^{1/4} (\cos(7\pi/4) + i\sin(7\pi/4))$$ 6. **Write the general solution:** Since roots are complex conjugates, the general solution is $$y = e^{\alpha x}(C_1 \cos(\beta x) + C_2 \sin(\beta x)) + e^{-\alpha x}(C_3 \cos(\beta x) + C_4 \sin(\beta x))$$ where $\alpha$ and $\beta$ are the real and imaginary parts of the roots. Here, $\alpha = 3^{1/4} \cos(\pi/4) = 3^{1/4} \frac{\sqrt{2}}{2}$ and $\beta = 3^{1/4} \sin(\pi/4) = 3^{1/4} \frac{\sqrt{2}}{2}$. 7. **Final answer:** $$\boxed{y = e^{3^{1/4} \frac{\sqrt{2}}{2} x} (C_1 \cos(3^{1/4} \frac{\sqrt{2}}{2} x) + C_2 \sin(3^{1/4} \frac{\sqrt{2}}{2} x)) + e^{-3^{1/4} \frac{\sqrt{2}}{2} x} (C_3 \cos(3^{1/4} \frac{\sqrt{2}}{2} x) + C_4 \sin(3^{1/4} \frac{\sqrt{2}}{2} x))}.$$