Question

Consider an LTI system whose input x(t) and output y(i) are related by the

differential equation
d/dt y(t) + 4 y(t) = x(t) (eq. (P2.17-1)).
The system also satisfies the condition of initial rest.

If X(t) = (e(⁻¹⁺³))u(t), what is y(t)?

Answer 1

The question involves solving a first-order linear differential equation for an LTI system with given input and initial rest conditions. The output y(t) can be found using integrating factors or Laplace transforms, resulting in a summed solution of a homogeneous decaying exponential and a particular solution influenced by the input.

Step-by-step explanation:

The student asked for the output y(t) of an LTI (Linear Time-Invariant) system given the input x(t) = e(-t+3)u(t), where u(t) is the unit step function, and the differential equation relating the input to the output is dy/dt + 4y(t) = x(t). This is a first-order linear differential equation, and since the system is initially at rest, we can assume y(0) = 0. To solve this, we can use the method of integrating factors or apply Laplace transforms taking into account the initial conditions. For an input that includes the exponential function and unit step function, we should look for a solution in the form of a decaying exponential especially considering the presence of a damping factor given by the coefficient '4' in the differential equation.

The solution to this type of differential equation typically involves finding the homogeneous solution that satisfies dy/dt + 4y(t) = 0 and the particular solution that satisfies dy/dt + 4y(t) = x(t). For t > 0, setting the left-hand side equal to zero provides the homogeneous solution, which would be a decaying exponential due to the negative coefficient of y. The particular solution considering the input e(-t+3)u(t) will involve finding a constant that when plugged into the equation along with the exponential will satisfy the equation. The total solution y(t) will then be the sum of the homogeneous solution and the particular solution.