## Applied Mathematics - 3 - May 2016

### Mechanical Engineering (Semester 3)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1(a)** Find Laplace transform of $\dfrac{\sin^2 2t}{t} $(5 marks)
**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 marks)
**1(c)** Find complex form of Fourier series

f(x) = e^{3x} in 0 < x < 3(5 marks)
**1(d)** Shoe that the function is analytic and find their derivative f(z) = ze^{x}.(5 marks)
**2(a)** Using Laplace transform solve: $ \dfrac{d^2y}{dt^2}+y=t\ \ y(0)=1\ \ y'(0)=0 $(6 marks)
**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 marks)
**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 marks)
**3(a)** Find the bi-linear transformation that maps points 0. 1, ∞ of the z plane into -5, -1, 3 of w plane.(6 marks)
**3(b)** By using Convolution theorem find inverse Laplace transform of $$\dfrac{1}{(s-2)^4 (s+3)}$$(6 marks)
**3(c)** Find the Fourier series of f(x)

f(x) = cosx -π<x<0 <br=""> sinx 0<x<π< a="">

</x<π<><>(8 marks)
**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 marks)
**4(b)** Evaluate and prove that $$\int ^{\infty}_0 e^{-\sqrt{2}t}\dfrac{\sin t\sinh t}{t}=\dfrac{\pi}{8}$$(6 marks)
**4(c)** Obtain Laurent's series for the function, $$f(z)=\dfrac{-7z-2}{z(z-2)(z+1)}\ \text{about}\ z=1$$(8 marks)
**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) = x^{2}(25-x^{2}) taking h=1 upto 3 seconds only by Bender Schmidt formula.(6 marks)
**5(b)** Construct an analytic function whose real part is $\dfrac{\sin 2x}{\cosh 2y+\cos 2x} $(6 marks)
**5(c)** Evaluate $ \int ^{\pi}_0 \dfrac{d\theta}{3+2\cos \theta}$(8 marks)
**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 marks)
**6(b)** Evaluate : $ \int ^{1+l}_0 Z^2 dz\ \text{along}$

(i) The line y = x

(ii) The parabola x = Y^{2}

Is the line integral independent of path? Explain.(6 marks)
**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 $(8 marks)