1. The problem is to write the state-space equations for the plant given by the differential equation $$\frac{d^2 y(t)}{dt^2} + 3 \frac{d y(t)}{dt} + y(t) = r(t)$$. 2. Define the state variables to convert the second-order differential equation into first-order equations: Let $$x_1(t) = y(t)$$ and $$x_2(t) = \frac{d y(t)}{dt}$$. 3. Express the derivatives of the state variables: $$\frac{d x_1(t)}{dt} = x_2(t)$$ 4. Rewrite the original differential equation to express $$\frac{d^2 y(t)}{dt^2}$$: $$\frac{d^2 y(t)}{dt^2} = r(t) - 3 \frac{d y(t)}{dt} - y(t) = r(t) - 3 x_2(t) - x_1(t)$$ 5. Therefore, the derivative of $$x_2(t)$$ is: $$\frac{d x_2(t)}{dt} = r(t) - 3 x_2(t) - x_1(t)$$ 6. The state-space equations are: $$\begin{cases} \dot{x}_1(t) = x_2(t) \\ \dot{x}_2(t) = -x_1(t) - 3 x_2(t) + r(t) \end{cases}$$ 7. The output equation, since $$y(t) = x_1(t)$$, is: $$y(t) = x_1(t)$$ Final answer: $$\boxed{\begin{cases} \dot{x}_1 = x_2 \\ \dot{x}_2 = -x_1 - 3 x_2 + r(t) \\ y = x_1 \end{cases}}$$