1. **Problem Statement:** Given the function: $$y(t) = e^{-0.266t} \left\{ C_1 \begin{bmatrix} 0.732 \\ 1 \end{bmatrix} \cos 7.96t + C_2 \begin{bmatrix} 0.732 \\ 1 \end{bmatrix} \sin 7.96t \right\} + e^{-3.73t} \left\{ C_3 \begin{bmatrix} -2.73 \\ 1 \end{bmatrix} \cos 14.92t + C_4 \begin{bmatrix} -2.73 \\ 1 \end{bmatrix} \sin 14.92t \right\}$$ We want to understand the shape and behavior of this function. 2. **Explanation of Terms:** - The function is a sum of two parts, each with an exponential decay term multiplied by oscillatory terms (cosine and sine). - The exponential terms $e^{-0.266t}$ and $e^{-3.73t}$ represent decay rates; the larger the exponent's absolute value, the faster the decay. - The cosine and sine terms oscillate with angular frequencies $7.96$ and $14.92$ respectively. - Each oscillatory term is multiplied by a constant vector: - $\begin{bmatrix} 0.732 \\ 1 \end{bmatrix}$ for the first part - $\begin{bmatrix} -2.73 \\ 1 \end{bmatrix}$ for the second part - Constants $C_1, C_2, C_3, C_4$ scale each component. 3. **Behavior Analysis:** - The first term decays slowly (rate $0.266$) and oscillates with frequency $7.96$. - The second term decays faster (rate $3.73$) and oscillates with higher frequency $14.92$. - Over time, the second term's contribution diminishes quickly due to faster decay. - The overall function is a vector-valued function with two components, each a combination of these decaying oscillations. 4. **Summary:** This function models a system with two modes of oscillation and decay, where the first mode persists longer and oscillates slower, and the second mode fades quickly with faster oscillations.