Question

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

d/dt y(t)+4y(t)=x(t) The system also satisfies the condition of initial rest.
If x(t)=e⁽⁻¹⁺³ʲ⁾ᵗ u(t), what is y(t) ?

Answer 1

To find y(t), solve the given differential equation with the given input x(t) by substituting and solving.

Step-by-step explanation:

To find the output y(t) given the input x(t) and the differential equation d/dt y(t) + 4y(t) = x(t) with initial rest condition, we need to solve the differential equation. First, we can rewrite the equation as d/dt y(t) = x(t) - 4y(t). Then, we can substitute the given input x(t) = e^(-1+3j)t u(t) into the equation.

By substituting and solving the differential equation, we can find that y(t) = c * e^(-4t) + (1/17)(e^(-1+3j)t - e^(-4t)), where c is the constant of integration determined from the initial rest condition.