Question

The input and output of a stable and causal LTI system are related by the differential equation

d²y(t)/dt² + 4 dy(t)/dt + 3y(t) = x(t). Find the impulse response of the system

Answer 1

To find the impulse response of the system represented by the given differential equation, we can use the Laplace transform. By taking the Laplace transform of both sides of the equation and solving for Y(s), we can find the impulse response h(t) by taking the inverse Laplace transform of Y(s).

Step-by-step explanation:

Given the differential equation d²y(t)/dt² + 4 dy(t)/dt + 3y(t) = x(t), we can find the impulse response of the system by using the Laplace transform. Taking the Laplace transform of both sides of the equation, we get s²Y(s) + 4sY(s) + 3Y(s) = X(s), where Y(s) and X(s) are the Laplace transforms of y(t) and x(t) respectively.

Solving for Y(s), we get Y(s) = X(s)/(s² + 4s + 3).

Now, to find the impulse response, we can take the inverse Laplace transform of Y(s). Therefore, the impulse response h(t) = L^-1{Y(s)} = L^-1{X(s)/(s² + 4s + 3)}.