MU Applied Mathematics - 3 - May 2014 Exam Question Paper

MU Mechanical Engineering (Semester 3)
Applied Mathematics - 3
May 2014

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 the inverse Laplace transform of

\frac{S^{2} + 5}{(S^{2} + 4 S + 13)^{2}}

$\dfrac {S^{2}+5}{(S^{2}+4S+13)^{2}}$

5 M

1 (b) IF V=3x²y+6xy-y³, show that the funcltion V is harmonic, find the corresponding analytic function.

5 M

1 (c) Evaluate

\int_{c} ¯ z d z

$\int_{c}{\bar{z}dz}$ where C is the upper half of the circle r=1

5 M

1 (d)

$Prove \ that \ F_{1}(x)=1, f_{2}(x)=x, f_{3}(x)=\dfrac {3x^{2}-1}{2} \\ are \ orthogonal \ over \ (-1,1)$

5 M

2 (a)

$Evaluate \ \int^{\infty}_{0}\dfrac {\cos at - \cos bt}{t}dt$

6 M

2 (b) Obtain complex form of fourier series f(x)=e^ax for in (-π, π)

6 M

2 (c) Using Crank-Nicholson simplified formula solve ,

$\dfrac {\partial^{2}u}{\partial x^{2}}- \dfrac {\partial u}{\partial t}=0 u(0,t)=0, u(4,t)=0, u(x,o)=\dfrac {x}{3} (16-x^2)$ for uij i=0,1,2,3,4, and j=0,1,2

8 M

3 (a)

$Evaluate \ \int_{c}\dfrac {\sin^{6}z}{ \left (z-\dfrac {\pi}{6}\right )^{3}}\ where \ C \ is \ |z|=1$

6 M

3 (b) Find the fourier expansion for f(x)=x-x²-1<x<1

6 M

3 (c) Determine the solution of one dimensional heat equation,

$\dfrac {\partial u}{\partial t}= C^{2} \dfrac {\partial^{2}u}{\partial x^{2}}$ under the boundary conditions u(0,t)=0 u(l,t)=0 and u(x,0)=x, (0<x<l), l being length of the rod.

8 M

4 (a) Find inverse Laplace transform by using convolution theorem,

$f(s)= \dfrac {s^{2}}{(s^{2}-a^{2})^{2}}$

6 M

4 (b) Find the image of the region bounded by x=0, x=2, y=0, y=2 in the Z plane under transformation W=(1+i)Z.

6 M

4 (c) Find all possible Laurent's expansion of the function

$f(z)= \dfrac {7z-2}{z(z-2)(z+1)} \ about \ Z=-1$

8 M

5 (a)

$Solve\ \dfrac {\partial^{2}u}{\partial x^{2}}-32 \dfrac {\partial u}{\partial t}=0$ by Bender-Schmidt method, subject to the conditions u(0,t)=0, u(x,0)=0, u(1,t)=t taking h=0.25, 0 <x <1

6 M

5 (b) Obtain half range sine series for f(x) when

$\begin {align*} f(x)&=x, \ \ &0<x<\dfrac {\pi}{2}\\&=\pi -x, \ &\dfrac {\pi}{2}<x<\pi\end{align*}$

6 M

5 (c)

$Evaluate \ \int^{\infty}_{-\infty}\dfrac {x^{2}dx}{(x^{2}+a^{2})(x^{2}+b^{2})}$ by using residues a>0, b>0

8 M

6 (a) Find the orthogonal trajectory of the family of curves x³y-xy³=c

6 M

6 (b) Obtain the fourier expansion of

$f(x)=\left (\dfrac {\pi - x}{2} \right )^{2}$ in the interval 0<x<2π, f(x+2π)=f(x) also deduce that

$\dfrac {\pi}{6}=\dfrac {1}{1^{2}}+\dfrac {1}{2^{2}}+\dfrac {1}{3^{2}}+......$

6 M

6 (c) Solve using Laplace transform (D²-3D+2)y=4 e^2t, with y(0)=-3 y'(0)=5

8 M

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