Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 17 Scheme 17MAT21 Engineering Mathematics II Question Paper

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CBCS SCHEME 17MAT21

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

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Engineering Mathematics - II

Time: 3 hrs.

Max. Marks: 100

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

Module-1

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1. a. Solve (D² +1)y = 3x² + 6x +12. (06 Marks)
   b. Solve (D³ +2D² + D)y = e^-x (07 Marks)
   c. By the method of undetermined coefficients, solve (D² + D- 2)y = x + sin x . (07 Marks)

OR

1. a. Solve (D² -6D + 9)y = 6e^3x +7e^2x. (06 Marks)
   

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   b. Solve (D³ - D)y= (2x +1)+4cosx . (07 Marks)
   c. By the method of variation of parameters, solve (D² +1)y = cosec x . (07 Marks)

Module-2

1. a. Solve x²y"-3xy' + 4y =1+ x². (06 Marks)
   b. Solve xyp² -(x² + y² )p + xy =0 . (07 Marks)
   

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   c. Solve (px -y)(py+ x) = a²p by taking x² = X and y² = Y. (07 Marks)

OR

1. a. Solve (2+ x)² y"+(2+ x)y' =sin(2log(2+ x)). (06 Marks)
   b. Solve yp² + (x - y)p- x=0 . (07 Marks)
   c. Obtain the general and singular solution of the equation sin px cosy = cospx sin y+ p. (07 Marks)

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Module-3

1. a. Form a partial differential equation by eliminating arbitrary function f(lx + my +nz)(x² + y² + z²) (06 Marks)
   b. Solve ?z/?x =xy subject to the conditions z = log(1+ y) when x = 1 and z = 0 when x = 0. (07 Marks)
   c. Derive an expression for the one dimensional wave equation. (07 Marks)

OR

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1. a. Form a partial differential equation z = f(y + 2x) + g(y-3x) (06 Marks)
   b. Solve ?²z/?x?y = z, given that when y = 0, z =e^x and ?z/?y = e^-x . (07 Marks)
   c. Find all possible solutions of heat equation u _t = c²u_xx by the method of separation of variables. (07 Marks)

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Module-4

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1. a. Evaluate ? r sin ? dr d? over the cardioids r = a(1-cos ?) above the initial line. (06 Marks)
   b. Evaluate ? 1/v(x) dz dx dy (07 Marks)
   c. Derive the relation between Beta and Gamma function as B(m, n) = G(m)G(n) / G(m + n) (07 Marks)

OR

1. a. Evaluate ? e^y² dydx by changing the order of integration. (06 Marks)
   

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   b. Find by double integration, the area lying between the parabola y² = 4x - 2 and the line y = x. (07 Marks)
   c. Show that ?₀^p/2 vcot ? d? = p/v2 (07 Marks)

Module-5

1. a. Find the Laplace transform of t cos 2t (06 Marks)
   b. Find the Laplace transform of f(t) = |sin ?t|, 0 c. Solve y" - 3y' + 2y = 2e^3t, y(0) = 0, y'(0) = -1 by using Laplace transforms. (07 Marks)

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OR

1. a. Find the inverse Laplace transforms of s/(s²+4)² +log(s+1/s) (06 Marks)
   b. By using convolution theorem, find L^-1{ 1 / (s² + a²)² } (07 Marks)
   c. Express f(t) = { sint, 0

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