Final answer:
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.