Final answer:
To find the Fourier series representation of the output, we need to solve the differential equation given and express the solution as a sum of sinusoidal functions. The general solution consists of a homogeneous solution and a particular solution. The Fourier series representation of the output is obtained by expressing the particular solution in terms of complex exponentials.
Step-by-step explanation:
To find the Fourier series representation of the output, we need to solve the differential equation given. First, let's find the homogeneous solution by assuming y(t) = e^(rt). Substituting this into the differential equation gives r^2 + 4r + 4 = 0, which has a double root of r = -2. Therefore, the homogeneous solution is y_h(t) = (c1 + c2t)e^(-2t).
Next, let's find a particular solution. Since the input signal is a sum of cosines and sines, we can assume a particular solution of the form y_p(t) = A*cos(4πt) + B*sin(4πt) + C*sin(6πt+π/4). Plugging this into the differential equation and equating like terms, we find that A = -1/4, B = 0, and C = 1/4. Therefore, the particular solution is y_p(t) = (-1/4)*cos(4πt) + (1/4)*sin(6πt+π/4).
The general solution is given by y(t) = y_h(t) + y_p(t). To find the Fourier series representation of y(t), we need to express it as a sum of sinusoidal functions. Using Euler's formula, we can rewrite the cosine and sine terms as complex exponentials. After simplifying, we find that the Fourier series representation of y(t) is: y(t) = (c1 + c2t)e^(-2t) + (-1/4)*(e^(4πit)+e^(-4πit))/2 + (1/4)*(e^(3πit + π/4)+e^(-3πit - π/4))/2.