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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

1. 0.2 + 0.8e⁻⁵t
2. 0.2 - 0.2e⁻⁵t
3. 0.8 + 0.2e⁻⁵t
4. 0.8 - 0.8e⁻⁵t

Correct answer: 0.2 + 0.8e⁻⁵t

Solution

The correct option is derived from solving the first-order linear differential equation using the method of integrating factors. The particular solution for a unit step input leads to a steady-state output of 0.2, while the homogeneous solution decays over time, resulting in the final expression of 0.2 + 0.8e⁻⁵t.

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