\( y ^ { \prime \prime } + y ^ { \prime } + y = 0 \)

This second-order linear homogeneous differential equation has a characteristic equation formed by substituting \( y = e^{rt} \), leading to \( r^2 + r + 1 = 0 \). Solving this using the quadratic formula gives \( r = \frac{-1 \pm \sqrt{-3}}{2} \), resulting in complex roots \( r = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i \). This indicates a solution involving oscillatory behavior with exponential decay: \( y(t) = e^{-\frac{1}{2}t}(C_1 \cos(\frac{\sqrt{3}}{2}t) + C_2 \sin(\frac{\sqrt{3}}{2}t)) \). To tackle this problem, you can start by calculating the roots of the characteristic equation correctly. A common mistake is overlooking the imaginary parts, which can lead to an incorrect understanding of the oscillatory nature of the solutions. Make sure you articulate the solution in terms of both real and imaginary components to fully capture the behavior of the system!