A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system. y(t) + 5y(t) = u(t) When y(0) = 1 and u(t) is a unit step function, y(t) is

The correct answer is $\boxed{\text{A) }0.2 + 0.8e^{-5t}}$.

The differential equation $y(t) + 5y(t) = u(t)$ is a first-order linear differential equation with constant coefficients. The general solution to this equation is $y(t) = C\cdot e^{-5t}$, where $C$ is an arbitrary constant.

When $y(0) = 1$, we have $C = 1$. Therefore, the solution to the differential equation is $y(t) = e^{-5t}$.

When $u(t)$ is a unit step function, the output of the system is $y(t) = 0.2 + 0.8e^{-5t}$.

Here is a brief explanation of each option:

• Option A: $0.2 + 0.8e^{-5t}$ is the correct answer. This is the solution to the differential equation $y(t) + 5y(t) = u(t)$ when $y(0) = 1$ and $u(t)$ is a unit step function.
• Option B: $0.2 – 0.2e^{-5t}$ is not the correct answer. This is the solution to the differential equation $y(t) + 5y(t) = 0$ when $y(0) = 1$.
• Option C: $0.8 + 0.2e^{-5t}$ is not the correct answer. This is the solution to the differential equation $y(t) + 5y(t) = 2$ when $y(0) = 1$.
• Option D: $0.8 – 0.8e^{-5t}$ is not the correct answer. This is the solution to the differential equation $y(t) + 5y(t) = 4$ when $y(0) = 1$.