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
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: ∂u∂x2−∂u∂t=0u(0,t)=0 u(4,t)u(x,0)=x3(16−x2)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(s−2)4(s+3)
6 M
3(c)
Find the Fourier series of f(x)
f(x) = cosx -π sinx 0
f(x) = cosx -π
8 M
4(a)
Find half range sine series for x sin x in (0, π) and hence deduce π28√2=112−132+152−172⋯
6 M
4(b)
Evaluate and prove that ∫∞0e−√2tsintsinhtt=π8
6 M
4(c)
Obtain Laurent's series for the function, f(z)=−7z−2z(z−2)(z+1) about z=1
8 M
5(a)
Solve : ∂2u∂x2−∂u∂t=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. ∂2y∂t2=c2∂2y∂x2
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.
(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)=(π−x2)2
in the interval 0 ≤ x ≤ 2π and f(x+2π) = f(x) and also deduce
(i) π212=112−122+132−142⋯
(ii) π490=114+124+134+144⋯
in the interval 0 ≤ x ≤ 2π and f(x+2π) = f(x) and also deduce
(i) π212=112−122+132−142⋯
(ii) π490=114+124+134+144⋯
8 M
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