# 1. A function is given by 𝑓(𝑥) = 1 − |𝑥| 𝑇 , where 𝑥 ∈ [−𝑇, 𝑇]. State whether the function is even or odd and hence find the Fourier series for this function. Qualified Writers
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This coursework constitutes 20% of the total module mark. The submission deadline is Monday, 16/01/2017, 12 pm. Please hand in your work in the undergraduate office in MECH G43. Make sure your work has your name and ID number written clearly on each sheet and all the sheets are stapled together. The work should be neatly presented. Credit will be given to well-presented work. Please solve a new question on a new page.

1. A function is given by 𝑓(𝑥) = 1 − |𝑥| 𝑇 , where 𝑥 ∈ [−𝑇, 𝑇]. State whether the function is even or odd and hence find the Fourier series for this function.

2. Find the Fourier series for the function 𝑓(𝑥) = 𝑥 3 − 2𝑥 2 , defined on −𝜋 < 𝑥 < 𝜋 (Hint: Use the following information to simplify integration. If 𝑂(𝑥) is odd function and 𝐸(𝑥) is even function then 𝑔(𝑥) = 𝑂(𝑥) ∗ 𝐸(𝑥) is an odd function and ∫ 𝑔(𝑥) 𝑇 −𝑇 = 0)

3. Find the Fourier transform of the function 𝑓(𝑡) = { 0 𝑡 < − 𝜋 2 ⁄ (4𝑡 + 1) − 𝜋 2 ⁄ ≤ 𝑡 ≤ 𝜋 2 ⁄ 0 𝜋 2 ⁄ < 𝑡 }

4. Using the definition of Laplace Transform, show that: L{ln(𝑥)} = − ( 𝛾 + ln(𝑠) 𝑠 ), Where, 𝛾 = − ∫ 𝑒 −𝑥 ln(𝑥)𝑑𝑥 ∞ 0

5. Using the formulae of Laplace transforms for derivatives, find the Laplace Transform of L{t sin at} and L{𝑡 cos 𝑎𝑡}.

6. Solve for 𝑥(𝑡) the following differential equation which has an impulse input, 𝑥̈+ 6𝑥̇ + 8𝑥 = 4. 𝛿(𝑡 − 5) 𝑥(0) = 0, 𝑥̇(0) = 3

7. Given that 𝐻(𝑡) is a Heaviside or step function, solve the following integro differential equation 𝑑𝑥 𝑑𝑡 + 6𝑥 + 9 ∫ 𝑥𝑑𝑡 𝑡 0 = 𝐻(𝑡), where 𝑥(0) = 0

8. Using the method of Laplace Transforms, solve the following initial valued problem 𝑦 ′ + 2𝑦 = 4𝑡𝑒 −2𝑡 , 𝑦(0) = −3

9. Assuming the current at 𝑡 = 0 is zero in the following circuit; find the equation for current 𝑖(𝑡) using the method of Laplace Transforms.

10. Solve the difference equation 𝑦[𝑘 + 3] − 6𝑦[𝑘 + 2] + 11𝑦[𝑘 + 1] − 6𝑦[𝑘] = 0 subject to 𝑦(0) = 0, 𝑦(1) = 2 and 𝑦(2) = 2 by z-Transform method.

1. A function is given by 𝑓(𝑥) = 1 − |𝑥| 𝑇 , where 𝑥 ∈ [−𝑇, 𝑇]. State whether the function is even or odd and hence find the Fourier series for this function.

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