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Consider a causal LTI system characterized by differential equation dy(t)/dt + (1/6)y(t) = 3x(t). The response of the system to the input x(t) = 3e^(-t/3)u(t), where u(t) denotes the unit step function, is

1. 9e^(-t/3)u(t).
2. 9e^(-t/6)u(t).
3. 9e^(-t/3)u(t) - 9e^(-t/6)u(t).
4. 54e^(-t/6)u(t) - 54e^(-t/3)u(t).

Correct answer: 54e^(-t/6)u(t) - 54e^(-t/3)u(t).

Solution

The correct option represents the system's response to the input by applying the principle of superposition and the system's impulse response, which results in a combination of exponential terms that reflect the system's dynamics and the input characteristics.

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