Question

Find the zero-input response of an LTIC system described by (D+5)y(t)= x(t) if the initial condition is y₀​(t)=5.

Letting y₀​(t)=1 and y₀​(t)=4, solve (D²+2D)y₀​(t)=0

Answer 1

To find the zero-input response of an LTIC system with a given initial condition, rearrange the equation, integrate both sides, solve for the constant of integration using the given initial condition, and substitute the constant back into the equation.

Step-by-step explanation:

To find the zero-input response of the given LTIC system (D+5)y(t)= x(t) with the initial condition y₀​(t)=5, we need to solve the differential equation with the initial condition. The given equation is a first-order differential equation, which can be solved using the integrating factor method.

1. First, rearrange the equation to isolate y(t): y(t) = (1/(D+5))x(t).
2. Next, integrate both sides of the equation to find the zero-input response of the system: ∫y(t)dt = ∫(1/(D+5))x(t)dt.
3. Using the given initial condition y₀​(t)=5, substitute it into the equation and solve for the constant of integration.
4. The final step is to substitute the constant of integration back into the equation to find the zero-input response of the system.

By following these steps, you can find the zero-input response of the LTIC system described by (D+5)y(t)= x(t) with the initial condition y₀​(t)=5.