5. (20%) Solve coupled differential equations,
  subjected to the constraints represent positive flow quantity in the domain is a positive constant. The following boundary conditions are also enforced,
  

(a) (10%) Let A be a square matrix of order n and A is normal. w is an arbitrary non-zero 1 X n column vector, proof that λ1is the largest eigenvalue corresponding to eigenvector where

(b) (10%) Let λ2 is an eigenvalue of A such that |λ2 -λ1| ＞|λp-λ1  whereλp is any eigenvalue of A other than λ2 and λ1. Describe a method to find λ2 and its associated eigenvector and prove that method in details.

(a) (10%) Calculate the Fourier transform of the function, You are supposed to show detailed calculations.

(b) (10%) Solve y from the differential equation under the condition that y(±∞) = 0

3. (20%) Three particles of identical mass m are connected by springs and attached to non-moving walls as shown in Fig. 1. All the springs are identical and have a spring constant k. The forces exerted on particles obey hooks law. Gravitation is neglected. Find the equation of motion for the three particles deviating from their equilibrium positions (x1, X2, x3 in Fig. 1), respectively and find the eigenfunctions and corresponding eigenvalues of the oscillator system.

(a) (10%) Evaluate the integral

(b) (10%) Evaluate the Fourier transform of the chirp functions,

(b) (6%), (7%) Please give their (Equations (2) and (3)) matching engineering (or physical) examples with discussion, respectively.

(a) (6%), (6%) Please first prove Equations (2) and (3), respectively

(d) (5%) Can one obtain identical Fourier series of any function g(t) by using 2T and T in Equation (1)? Explain your reasons within 30 words.

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