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MU Mechanical Engineering (Semester 3)
Applied Mathematics - 3
May 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary


1(a) Find Laplace transform of sin22tt
5 M
1(b) Find the orthogonal trajectory of the family of curves e-x cosy + xy = α where α is a real constant in the xy plane.
5 M
1(c) Find complex form of Fourier series
f(x) = e3x in 0 < x < 3
5 M
1(d) Shoe that the function is analytic and find their derivative f(z) = zex.
5 M

2(a) Using Laplace transform solve: d2ydt2+y=t  y(0)=1  y(0)=0
6 M
2(b) Using Crank Nicholson method Solve: ux2ut=0u(0,t)=0  u(4,t)u(x,0)=x3(16x2)find uij
6 M
2(c) Shoe that the set of functions 1,sinπxL,cosπxL,sin2πxL,cos2πxL form an orthogonal set in (-L, L) and construct an orthonormal set.
8 M

3(a) Find the bi-linear transformation that maps points 0. 1, ∞ of the z plane into -5, -1, 3 of w plane.
6 M
3(b) By using Convolution theorem find inverse Laplace transform of 1(s2)4(s+3)
6 M
3(c) Find the Fourier series of f(x)
f(x) = cosx     -π     sinx     0
8 M

4(a) Find half range sine series for x sin x in (0, π) and hence deduce π282=112132+152172
6 M
4(b) Evaluate and prove that 0e2tsintsinhtt=π8
6 M
4(c) Obtain Laurent's series for the function, f(z)=7z2z(z2)(z+1) about z=1
8 M

5(a) Solve : 2ux2ut=0 subject to the conditions u(0, t) = 0, u(5, t) = 0, u(x, 0) = x2(25-x2) taking h=1 upto 3 seconds only by Bender Schmidt formula.
6 M
5(b) Construct an analytic function whose real part is sin2xcosh2y+cos2x
6 M
5(c) Evaluate π0dθ3+2cosθ
8 M

6(a) An elastic string is stretched between two points at a distance l apart. In its equilibrium position a point at a distance a(a < l) from one end is displaced through a distance b transversely and then released from position. Obtain y(x, t) the Vertical displacement if y satisfies the equation. 2yt2=c22yx2
6 M
6(b) Evaluate : 1+l0Z2dz along
(i) The line y = x
(ii) The parabola x = Y2
Is the line integral independent of path? Explain.
6 M
6(c) Find Fourier expansion of f(x)=(πx2)2
in the interval 0 ≤ x ≤ 2π and f(x+2π) = f(x) and also deduce
(i) π212=112122+132142
(ii) π490=114+124+134+144
8 M



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