STUPIDSID
1(a)
Find Laplace transform of \(\dfrac{\sin^2 2t}{t} \)
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: \( \dfrac{d^2y}{dt^2}+y=t\ \ y(0)=1\ \ y'(0)=0 \)
6 M
2(b)
Using Crank Nicholson method \[\text{Solve:}\ \dfrac{\partial ^u}{\partial x^2}-\dfrac{\partial u}{\partial t}=0\\\\ u(0,t)=0\ \ u(4,t)\\\\ u(x,0)=\dfrac{x}{3}(16-x^2)\text{find}\ u_{ij}\]
6 M
2(c)
Shoe that the set of functions \( 1,\sin\dfrac{\pi x}{L},\cos\dfrac{\pi x}{L},\sin\dfrac{2\pi x}{L},\cos\dfrac{2\pi x}{L}\cdots\) 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 \[\dfrac{1}{(s-2)^4 (s+3)}\]
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 \[\dfrac{\pi^2}{8\sqrt{2}}=\dfrac{1}{1^2}-\dfrac{1}{3^2}+\dfrac{1}{5^2}-\dfrac{1}{7^2}\cdots\]
6 M
4(b)
Evaluate and prove that \[\int ^{\infty}_0 e^{-\sqrt{2}t}\dfrac{\sin t\sinh t}{t}=\dfrac{\pi}{8}\]
6 M
4(c)
Obtain Laurent's series for the function, \[f(z)=\dfrac{-7z-2}{z(z-2)(z+1)}\ \text{about}\ z=1\]
8 M
5(a)
Solve : \(\dfrac{\partial ^2 u}{\partial x^2}-\dfrac{\partial u}{\partial t}=0 \) 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 \(\dfrac{\sin 2x}{\cosh 2y+\cos 2x} \)
6 M
5(c)
Evaluate \( \int ^{\pi}_0 \dfrac{d\theta}{3+2\cos \theta}\)
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. \[\dfrac{\partial ^2 y}{\partial t^2}=c^2\dfrac{\partial ^2y}{\partial x^2}\]
6 M
6(b)
Evaluate : \( \int ^{1+l}_0 Z^2 dz\ \text{along}\)
(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 \[f(x)=\left ( \dfrac{\pi-x}{2} \right )^2\]
in the interval 0 ≤ x ≤ 2π and f(x+2π) = f(x) and also deduce
\((i)\ \frac{\pi^2}{12}=\dfrac{1}{1^2}-\dfrac{1}{2^2}+\dfrac{1}{3^2}-\dfrac{1}{4^2}\cdots \)
\((ii)\ \frac{\pi^4}{90}=\dfrac{1}{1^4}+\dfrac{1}{2^4}+\dfrac{1}{3^4}+\dfrac{1}{4^4}\cdots \)
in the interval 0 ≤ x ≤ 2π and f(x+2π) = f(x) and also deduce
\((i)\ \frac{\pi^2}{12}=\dfrac{1}{1^2}-\dfrac{1}{2^2}+\dfrac{1}{3^2}-\dfrac{1}{4^2}\cdots \)
\((ii)\ \frac{\pi^4}{90}=\dfrac{1}{1^4}+\dfrac{1}{2^4}+\dfrac{1}{3^4}+\dfrac{1}{4^4}\cdots \)
8 M
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