Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 15 Scheme 15MAT21 Engineering Mathematics II Question Paper

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CBCS SCHEME

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15MAT21

Second Semester B.E. Degree Examination, June/July 2019

Engineering Mathematics II

Time: 3 hrs.

Max. Marks: 80

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

Module-1

1. 1. Solve (D²-4D + 4)y = e^2x + cos2x +4 by inverse differential operator method. (06 Marks)
   2. Solve d²y/dx² - 2dy/dx + 5y = e^2x sin x by inverse differential operator method. (05 Marks)
   3. Using the method of undetermined coefficients, solve y'' –3y' + 2y = x + e^x. (05 Marks)

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OR

1. 1. Solve d²y/dx² - 2dy/dx + y = x e^x sin x by inverse differential operator method. (06 Marks)
   2. Solve (D³ - 6D² + 11D – 6)y = e^2x + x by inverse differential operator method. (05 Marks)
   3. Solve y" – 2y' +y = e^x/x by method of variation of parameters. (05 Marks)

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Module-2

1. 1. Solve (2x-1)² d²y/dx² + (2x-1)dy/dx - 2y=8x² -2x +3 . (06 Marks)
   2. Solve xy(d²y/dx² + dy/dx ) + xy = 0 (05 Marks)
   3. Solve x²(y -px)- p²y = 0 by reducing into Clairaut's form and using the substation X = x² and Y = y². (05 Marks)

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OR

1. 1. Solve x²y"- xy'+ 2y = x sin(log x). (06 Marks)
   2. Obtain the general solution of the differential equation p² + 4x³p-12x⁴y = 0. (05 Marks)
   3. Obtain the general and singular solution of y = 2px + p²y. (05 Marks)

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Module-3

1. 1. Form the partial differential equation by eliminating the arbitrary function from the relation Z = y f(x) + x g(y). (06 Marks)
   2. Solve ?²z/?x?y = x sin y for which ?z/?x = -2sin y when x = 0 and z = 0 when y is an odd multiple of p/2. (05 Marks)
   3. Derive one dimensional wave equation ?²u/?t² = c²?²u/?x² (05 Marks)

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OR

1. 1. Form a partial differential by eliminating the arbitrary function (I) from the relation (x²+y²+z²,z²-2xy) = 0. (06 Marks)
   2. Solve ?z/?x + 4z = 0, given that when x = 0, z = e^-y and ?z/?x = 2. (05 Marks)
   3. Determine the solution of the heat equation ?u/?t = c²?²u/?x² by the method of separation of variables for the constant K is positive. (05 Marks)

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Module-4

1. 1. Evaluate ? (xy + e^x)dydx, limits are from 0 to 1 and 0 to x. (06 Marks)
   2. Evaluate ? dydx, limits are from 0 to 4a and x²/4a to 2vax by changing the order of integration. (05 Marks)
   3. Obtain the relation between the beta and gamma function in the form ß(m,n)= G(m)G(n) / G(m+n). (05 Marks)

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OR

1. 1. Evaluate ? e^-(x2+y2) dxdy by changing into polar coordinates. Limits are from 0 to 8 and 0 to 8. (06 Marks)
   2. Evaluate ? e^-zyⁿdzdydx. (05 Marks)
   3. Using beta and gamma function, prove that ?₀^p/2 sin^m? cosⁿ? d? = G((m+1)/2)G((n+1)/2) / 2G((m+n+2)/2). (05 Marks)

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Module-5

1. 1. Find L{ t / v(t² + a²) } (06 Marks)
   2. If f(t + 2p) = f(t), then prove that L{f(t)} = coth(ps/2) ?₀^2p e^-stf(t) dt (05 Marks)
   3. Find L^-1{ s / (s² ± a²)² } using convolution theorem. (05 Marks)

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OR

1. 1. Express f(t) = 0 < t < 1
      t 1 < t < 2
      2 t > 2 in term of unit step function and hence find its Laplace transform. (06 Marks)

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   2. Find L^-1{ (s+5) / (s²-6s+13) } (05 Marks)
   3. Employ the Laplace transform to solve the differential equation y"(t)+ 4y'(t)+4y(t)= e^-t with the initial condition y(0) = 0 and y'(0) = 0. (05 Marks)

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