Question

Consider the LTIC system described by the differential equation:

y¨(t) + 9y˙(t) + 20y(t) = f(t)
Assume f(t) = e⁻³ᵗu(t), y(0⁻) = 1, and y˙(0⁻) = 0. Find the complete response.

Answer 1

The complete response of the given LTIC system is y(t) = {4} / {3}e^{-4t} - {1} / {3}e^{-5t} + {1} / {3}e^{-3t}u(t) .

Step-by-step explanation:

To find the complete response of the linear time-invariant continuous (LTIC) system described by the given differential equation, we can follow these steps:

1. Find the homogeneous solution (natural response) of the system.

2. Find the particular solution (forced response) due to the input function f(t) .

3. Combine the homogeneous and particular solutions to obtain the complete response.

The differential equation is given by:

y''(t) + 9y'(t) + 20y(t) = f(t)

where f(t) = e^{-3t}u(t) .

Step 1: Homogeneous Solution

The homogeneous solution is obtained by setting the input f(t) to zero:

y''_h(t) + 9y'_h(t) + 20y_h(t) = 0

The characteristic equation is given by:

s^2 + 9s + 20 = 0

Solving this quadratic equation, we find the roots s_1 and s_2 :

s_1 = -4, quad s_2 = -5

The homogeneous solution is then given by:

y_h(t) = A e^{-4t} + B e^{-5t}

Step 2: Particular Solution

To find the particular solution, we assume a solution of the form y_p(t) = Ce^{-3t}u(t) and substitute it into the differential equation:

Ce^{-3t})'' + 9(Ce^{-3t})' + 20Ce^{-3t} = e^{-3t}u(t)

Solving for C , we find C = {1} / {3} .

Therefore, the particular solution is:

y_p(t) = {1} / {3}e^{-3t}u(t)

Step 3: Complete Response

The complete response is the sum of the homogeneous and particular solutions:

y(t) = y_h(t) + y_p(t)

y(t) = A e^{-4t} + B e^{-5t} + {1} / {3}e^{-3t}u(t)

Now, use the initial conditions y(0^-) = 1 and y'(0^-) = 0 to find the values of A and B.

y(0^-) = A + B + {1} / {3} = 1

y'(0^-) = -4A - 5B = 0

Solving these equations will give you the values of A and B. Once you have these constants, you can write the complete response for the given system.