Question

A continuous-time system that is initially at rest is described by: \[ \frac{dy(t)}{dt} + 3y(t) = 2x(t), \] where \(x(t)\) is the input voltage and \(y(t)\) is the output voltage. The impulse response of the system is:

• A. \(3e^{-2t}\)
• B. \(\tfrac{1}{3} e^{-2t}u(t)\)
• C. \(2e^{-3t}u(t)\)
• D. \(2e^{-3t}\)

Hint:

The impulse response is the inverse Laplace of the transfer function \(H(s)\). Always apply unit step \(u(t)\) to enforce causality.

Answer 1

The Correct Option is C

Step 1: System description.
Differential equation: \[ \frac{dy}{dt} + 3y = 2x(t) \] Taking Laplace transform (zero initial conditions): \[ sY(s) + 3Y(s) = 2X(s) \Rightarrow Y(s) = \frac{2}{s+3} X(s) \]

Step 2: Transfer function.
\[ H(s) = \frac{Y(s)}{X(s)} = \frac{2}{s+3} \]

Step 3: Impulse response.
Impulse response \(h(t)\) is inverse Laplace of \(H(s)\): \[ h(t) = \mathcal{L}^{-1}\left\{\frac{2}{s+3}\right\} = 2 e^{-3t} u(t) \]

Final Answer: \[ \boxed{2e^{-3t}u(t)} \]