Download AKTU B-Tech 2nd Sem 2016-17 EAS203 Engineering Mathematics II Question Paper

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B.TECH.

THEORY EXAMINATION (SEM–II) 2016-17

ENGINEERING MATHEMATICS - II

Max. Marks : 100

Note: Be precise in your answer. In case of numerical problem assume data wherever not provided.

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SECTION - A

1. Explain the following: 10 x 2 = 20

1. (a) Show that the differential equation y dx – 2x dy = 0 represents a family of parabolas.
2. (b) Classify the partial differential equation (1 - x²) d²z/dx² – 2xy d²z/dy?x + (1 - y²) d²z/dy² = 2z
3. (c) Find the particular integral of (D - a)²y = e^ax f''(x).

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4. (d) Write the Dirichlet's conditions for Fourier series.
5. (e) Prove that J'(x) = -J₁(x).
6. (f) Prove that L [e^atf(t)] = F(s – a)
7. (g) Find the Laplace transform of f(t) = sin at / t
8. (h) Write one and two dimensional wave equations.

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9. (i) Find the constant term when f(x) = |x| is expanded in Fourier series in the interval (-2, 2).
10. (j) Write the generating function for Legendre polynomial P(x).

SECTION - B

Attempt any five of the following questions: 5 x 10 = 50

1. 2. (a) Solve the differential equation (D² + 2D + 2)y = e^-xsec³x,

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2. (b) Prove that (n + 1)P_n+1(x) = (2n + 1)xP_n(x) – nP_n-1(x), where P_n(x) is the Legendre's function.
3. (c) Find the series solution of the differential equation 2x² d²y/dx² + x dy/dx + x(x + 1)y = 0.
4. (d) Using Laplace transform, solve the differential equation d²y/dt² + 9y = cos 2t ; y(0) = 1, y'(0) = -1.
5. (e) Obtain the Fourier series of the function, f(t) = t, -p < t < 0
   = -t , 0 < t < p.
   

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   Hence, deduce that 1/1² + 1/3² + 1/5² + ... = p²/8
6. (f) Solve d²u/dx² + d²u/dy² = 0 under the conditions u(0, y) = 0, u(l, y) = 0, u(x, 0) = 0 and u(x, a) = sin(npx/l)
7. (g) Solve the partial differential equation: (D³ – 4D²D' + 5DD'² – 2D'³)z = e^y+2x + v(y + x)
8. (h) Using convolution theorem find L^-1 [1/((s+1)(s²+1))]

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Attempt any two of the following questions: 2 x 15 = 30

1. 3. (a) Solve the differential equation (D²-2D+1)y = e^x sin x
2. (b) Solve the equation by Laplace transform method: dy/dt + 2y + ?y dt = sin t, y(0) = 1.
3. (c) Solve the partial differential equation (y² + z²) p – xyq + zx = 0, where p = ?z/?x & q = ?z/?y

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1. 4. (a) Find the Laplace transform of (cos at - cos bt) / t
2. (b) Express f(x) = 4x³ – 2x² – 3x + 8 in terms of Legendre's polynomial.
3. (c) Expand f(x) = 2x – 1 as a cosine series in 0 < x < 2.

1. 5. (a) Show that J₃(x) = (2/x - 1)J₁(x) - J₀(x).
2. (b) Solve ?z/?x + 3 ?z/?y + 5z = 0; z(0,y) = 2e^-y by the method of separation of variables.

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3. (c) A tightly stretched string with fixed end x = 0 and x = l is initially in a position given by y = a sin(px/l). If it is released from rest from this position, find the displacement y(x, t).