VASAVI COLLEGE OF ENGINEERING (Autonomous), HYDERABAD

B.E. II Year I-Semester Examinations, December - 2017

Engineering Mathematics - III

(Common to ME, EEE & ECE)

Time: 3 hours Max. Marks: 70

1. Note: Answer ALL questions in Part-A and any FIVE questions from Part-B

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Part-A (10 x 2 = 20 Marks)

1. 1. Define periodic function and give an example.
   2. State Dirichlet’s conditions for a Fourier series expansion.
   3. Define Fourier transform and its inverse transform.
   4. State convolution theorem on Fourier transforms.

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   5. Write the formula for the coefficient b_n in the Fourier series of f(x) in (c, c+2p).
   6. Form a partial differential equation by eliminating arbitrary function from z = f(x² - y²).
   7. Solve (D² – 4D + 4)y = 0.
   8. Solve (p – q) = z.
   9. Write the one-dimensional heat equation.

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   10. Write all possible solutions of one-dimensional wave equation.

Part-B (5 x 10 = 50 Marks)

1. Obtain the Fourier series for the function f(x) = x, -p < x < p.

2. Find the Fourier cosine transform of f(x) = e^-ax, x > 0, a > 0. Hence evaluate .

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3. Solve the partial differential equation (x² – yz)p + (y² – zx)q = (z² – xy).

4. Solve the differential equation (D² – 3D + 2)y = x + sinx.

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5. A tightly stretched string with fixed end points x = 0 and x = l is initially in a shape given by f(x) = kx(l – x), where k is a constant and then released from rest. Find the displacement u(x, t) at any point of the string at any time t.

6. Solve under the conditions u(0, t) = 0, u(l, t) = 0, u(x, 0) = x, 0 < x < l.

7. 1. Find the Fourier series to represent the function f(x) = x², -p < x < p.

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   2. Solve (D² – 2D + 5)y = e^x sinx.

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