Let's solve the problem step by step as requested. --- ### **Step 1: Write the Unforced ODE and Find the Characteristic Roots** Given the unforced system (\(u = 0\)), the ODE becomes: \[ \ddot{y} + 4\dot{y} + 8y = 0 \] This is a second-order linear homogeneous ODE. Assume a solution of the form \( y(t) = e^{\lambda t} \): \[ \lambda^2 e^{\lambda t} + 4\lambda e^{\lambda t} + 8 e^{\lambda t} = 0 \] \[ e^{\lambda t} (\lambda^2 + 4\lambda + 8) = 0 \] So, the **characteristic equation** is: \[ \lambda^2 + 4\lambda + 8 = 0 \] Solve for \(\lambda\): \[ \lambda_{1,2} = \frac{-4 \pm \sqrt{16 - 32}}{2} = \frac{-4 \pm \sqrt{-16}}{2} = \frac{-4 \pm 4i}{2} = -2 \pm 2i \] **Explanation**: The characteristic roots are complex conjugates: \(\lambda_1 = -2 + 2i\), \(\lambda_2 = -2 - 2i\). --- ### **Step 2: Find the Solution Using Initial Conditions** General solution: \[ y(t) = A e^{(-2 + 2i)t} + B e^{(-2 - 2i)t} \] Or, using Euler's formula (\(e^{i\theta} = \cos\theta + i\sin\theta\)), rewrite as: \[ y(t) = e^{-2t} \left[ C_1 \cos(2t) + C_2 \sin(2t) \right] \] Let's use the initial conditions: - \(y(0^-) = -1\) - \(\dot{y}(0^-) = 1\) At \(t = 0\): \[ y(0) = e^0 [C_1 \cdot 1 + C_2 \cdot 0] = C_1 = -1 \] Compute the derivative: \[ \dot{y}(t) = \frac{d}{dt} \left( e^{-2t} (C_1 \cos 2t + C_2 \sin 2t) \right) \] Apply product rule: \[ \dot{y}(t) = -2 e^{-2t} (C_1 \cos 2t + C_2 \sin 2t) + e^{-2t} [ -2C_1 \sin 2t + 2C_2 \cos 2t ] \] At \(t=0\): \[ \dot{y}(0) = -2 (C_1 \cdot 1 + C_2 \cdot 0) + [ -2C_1 \cdot 0 + 2C_2 \cdot 1 ] = -2C_1 + 2C_2 \] Set equal to initial condition: \[ -2C_1 + 2C_2 = 1 \] Recall \(C_1 = -1\): \[ -2(-1) + 2C_2 = 1 \implies 2 + 2C_2 = 1 \implies 2C_2 = -1 \implies C_2 = -\frac{1}{2} \] So the solution is: \[ \boxed{ y(t) = e^{-2t} \left[ -\cos(2t) - \frac{1}{2}\sin(2t) \right] } \] --- ### **Final Answer** \[ \boxed{ y(t) = e^{-2t} \left[ -\cos(2t) - \frac{1}{2}\sin(2t) \right], \quad t \geq 0^- } \] **Explanations after each step:** - **Step 1:** Found the characteristic roots of the ODE, which are complex conjugates indicating a damped oscillatory response. - **Step 2:** Used the initial conditions to solve for the constants in the general solution, resulting in the explicit solution above.