MEC702_Assignment_2 - MEC702 Assignment 2 This is an...

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Unformatted text preview: MEC702 Assignment 2 This is an individual assignment, so it requires individual submission. The date of submission is 14 26 12 . Any submission later than this time will incur a penalty of 20% of the marked score per day unless valid reason (with evidence) is provided. • The total marks for the assignment is 50 and contributes 5% to the students coursework. • The mode of submission is either by handing in a written solution sheet or emailing the scanned assignment to the lecturer/tutors email. • All workings to the solutions should be provided in the solution sheet which is submitted. • All solutions should be written in blue or black ink (no pencil or red ink allowed) and all writing should be easily legible. • Plagarism is strictly prohibited and the student will be penalized accordingly. The lecturer holds the right to question any student about any solution in the assignment and ask the student to reproduce any portion of the solution at any moment. • Week of October, to be submitted by 1 am , Monday, th Exercise. 1. Find the general solution of the following systems. Show details. a.) y10 = 8y1 − y2 y20 = y1 + 10y2 b.) y10 = 2y1 + y3 y20 = 3y2 + 4y3 y30 = y3 2. (4 + 4 marks) Determine the type and stability of the critical point of the systems. a.) y10 = −3y1 + 2y2 y20 = −2y1 − 3y2 b.) y10 = −y1 + 2y2 y20 = −2y1 − y2 (3 + 3 marks) 3. Solve the IVPs by the Laplace transform. . (a) y 00 − 6y 0 + 5y = 29 cos 2t, y(0) = 3.2, y 0 (0) = 6.2 (b) y 00 + 3y 0 + 2.25y = 9t + 64, y(0) = 1, y 0 (0) = 31.5 (c) y 00 − 2y 0 − 3y = 0, y(4) = −3, y 0 (4) = −17. . [Hint: Part (c) is a shifted data problem. Refer to Example 6, Section 6.2 of the text book for the method to solve it] (4 + 4 + 4 marks) 2 4. 5. 6. Find the Fourier series of the given function f (x), which assumed to have the period 2π. Show the details of your work. (5 marks) Find (a) the Fourier cosine series expansion, (b) the Fourier sine series expansion for the function below. Find the Fourier integral for the function f (x). (4 + 4 marks) ( ex , 0 < x < 1 f (x) = . 0, x > 1 7. (5 marks) Show that for a signal of eight sample values, we have π 1 w = e− 4 i = √ (1 − i). 2 Check by squaring. (Read through Discrete and Fast Fourier Transform notes in the book) (6 marks) 3 ...
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