Question

This problem is related to Problems 9.33 - 9.38 in the text. We have solved differential equations using the method of undetermined coefficients (Chapter 7) and Laplace transforms (Chapter 8). We can use Fourier series to find the particular solution of an arbitrary order differential equation - as long as the driving function is periodic and can be represented by a Fourier series. In the problem description and answers, all numerical angles(phases) should be given in radian angles (not degrees). Given the differential equation y" + 14y' + 65y = 7cos(11t+ 0.785398) + 6cos(14t – 0.523599). = a. Write the fundamental frequency that you will use for this system. Wo - 1 help (numbers) b. Write the transfer function for this system as a function of n. (See page 530 of the text.) H= H(n) 1 help (formulas) = c. Find the particular solution, yp(t). No step function in this answer. Note the accuracy is 0.01% relative error. If the numbers of small, that means you need more decimal places to get the right answer. help (formulas) Yp(t) =

Answer 1

To solve for the fundamental frequency, transfer function, and particular solution of the given differential equation, one must consider the frequencies present in the periodic driving function, construct the transfer function based on the characteristic equation, and find the sum of responses to each periodic term using undetermined coefficients or Fourier series.

Step-by-step explanation:

To solve the given differential equation y" + 14y' + 65y = 7cos(11t+ 0.785398) + 6cos(14t – 0.523599), we need to:

a. Fundamental Frequency

The fundamental frequency ω0 for the system is determined by the periodic components of the driving function. In this case, we have cosine functions with frequencies of 11 and 14 rad/s. The smallest common multiple of these frequencies would typically be used for the Fourier series representation.

b. Transfer Function H(n)

The transfer function for the system as a function of n can be written as H(n) = 1 / (n² + 14n + 65), which corresponds to the characteristic equation derived from the homogeneous part of the differential equation.

c. Particular Solution yp(t)

Since the particular solution yp requires solving an equation involving trigonometric functions, we look for a solution of the form Acos(ωt + φ), where A and φ are the amplitude and phase shift, respectively. Then, we use the transfer function to account for the response of the system to each cosine term of the driving function separately, summing the responses together to find yp(t).