Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2018 Jan 2017 Dec I5MAT21 Engineering Mathematics II Question Paper

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I5MAT21

Second Semester B.E. Degree Examination, Dec.2017/Jan.2018

Engineering Mathematics - II

Time: 3 hrs.

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Max. Marks: 80

Note: Answer any FIVE full questions, choosing ONE full question from each module.

Module-1

1. a. Solve (d³y/dx³) - 2(d²y/dx²) + 4(dy/dx) = sinh(2x + 3) by inverse differential operator method. (05 Marks)
2. b. Solve (d²y/dx²) - 3(dy/dx) + 2y = xe^3x + sin 2x by inverse differential operator method. (05 Marks)

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3. c. Solve (d²y/dx²) + 4y = tan 2x by the method of variation of parameters. (06 Marks)

OR

1. a. Solve y" - 2y' + y = x cos x by inverse differential operator method. (05 Marks)
2. b. Solve (d²y/dx²) + 4y = x² + x + log 2 by inverse differential operator method. (05 Marks)
3. c. Solve (d²y/dx²) + 2(dy/dx) - 4y = 2x² - 3e^-x by the method of undetermined coefficients. (06 Marks)

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Module-2

1. a. Solve x³(d³y/dx³) + 3x²(d²y/dx²) + x(dy/dx) = x² + log x. (05 Marks)
2. b. Solve y = 2px + tan^-1(y p³). (05 Marks)
3. c. Solve (x²+y²)dy/dx = xy. (06 Marks)

OR

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1. a. Solve (2x + 5)² y" - 6(2x + 5)y' + 8y = 6x. (05 Marks)
2. b. Solve y = 2px + v(1 + p²). (05 Marks)
3. c. Solve the equation : (px - y)(py + x) = a²p by reducing into Clairaut's form, taking the substitution X = x², Y = y² (06 Marks)

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Module-3

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1. a. Obtain the partial differential equation by eliminating the arbitrary functions f and g: z = yf(x/y) + xg(y). (05 Marks)
2. b. Solve ?z/?x = xy subject to the conditions z = log(1 + y) when x = 1 and z = 0 when x = 0. (05 Marks)
3. c. Derive one dimensional heat equation in the form ?u/?t = c² ?²u/?x². (06 Marks)

OR

1. a. Obtain the partial differential equation given f(z, x² - y²) = 0. (05 Marks)

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2. b. Solve ?²z/?x² - 3 ?z/?x + 4z = 0 subject to the conditions that z = 1 and ?z/?x = -1 when x = 0. (05 Marks)
3. c. Obtain the solution of one dimensional wave equation ?²u/?t² = c² ?²u/?x² by the method separation of variables for the positive constant. (06 Marks)

Module-4

1. a. Evaluate I = ?₀² ?₀^x ?₀^x+y xyz dz dy dx. (05 Marks)
2. b. Find the area of the ellipse x²/a² + y²/b² = 1 by double integration. (05 Marks)

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3. c. Derive the relation between beta and gamma function as ß(m, n) = G(m)G(n) / G(m + n). (06 Marks)

OR

1. a. Evaluate ?₀^a ?₀^v(a2-y2) x dx dy by changing the order of integration. (05 Marks)
2. b. Evaluate ?₀^a ?₀^v(a2-x2) v(x² + y²) dx dy by changing into polar co-ordinates. (05 Marks)
3. c. Evaluate ?₀^p/2 v(sin ?) d? by using Beta-Gamma functions. (06 Marks)

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Module-5

1. a. Find the Laplace transform of (cos 2t - cos 3t)/t + t sint. (05 Marks)
2. b. Express the function f(t) = { t, 0p } in terms of unit step function and hence find its Laplace transform. (05 Marks)
3. c. Solve y" + 6y' + 9y = e^-3t subject to the conditions, y(0) = 0 = y'(0) by using Laplace transform. (06 Marks)

OR

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1. a. Find the inverse Laplace transform of (7s + 4) / (s² + 4s + 9). (05 Marks)
2. b. Find the Laplace transform of the full wave rectifier f(t) = E sin ?t, 0 < t < p/? having period p/?. (05 Marks)
3. c. Obtain the inverse Laplace transform of the function 1 / ((s-1)(s²+1)) by using convolution theorem. (06 Marks)

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