Question

Let a causal LTI system be governed by the following differential equation \[ y(t) + \frac{1}{4} \frac{dy}{dt} = 2x(t), \] where \( x(t) \) and \( y(t) \) are the input and output respectively. Its impulse response is

• A. ( 2e^{-\frac{1}{4}t}u(t) \)
• B. ( 2e^{-4t}u(t) \)
• C. ( 8e^{-\frac{1}{4}t}u(t) \)
• D. ( 8e^{-4t}u(t) \)

Hint:

For an LTI system described by a first-order differential equation, the impulse response is found by taking the inverse Laplace transform of the transfer function.

Answer 1

The Correct Option is A

Step 1: Understanding the System.
This is a first-order linear differential equation. To find the impulse response \( h(t) \), we need to solve for the system's response when the input is a Dirac delta function \( \delta(t) \). Step 2: Solving the Differential Equation.
To find the impulse response, we first take the Laplace transform of the given equation. The Laplace transform of the equation is: \[ Y(s) + \frac{1}{4}sY(s) = 2X(s) \] Solving for the transfer function \( H(s) = \frac{Y(s)}{X(s)} \), we get: \[ H(s) = \frac{2}{s + \frac{1}{4}} \] Taking the inverse Laplace transform of \( H(s) \), we get the impulse response: \[ h(t) = 2e^{-\frac{1}{4}t}u(t) \] Step 3: Conclusion.
The correct answer is (A) \( 2e^{-\frac{1}{4}t}u(t) \), which is the impulse response of the system.