MU Applied Mathematics - 3 - May 2014 Exam Question Paper

MU Civil 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} \bar{z} d z

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

5 M

1 (d)

P r o v e t h a t F_{1} (x) = 1, f_{2} (x) = x, f_{3} (x) = \frac{3 x^{2} - 1}{2} a r e o r t h o g o n a l o v e r (- 1, 1)

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

5 M

2 (a)

E v a l u a t e \int_{0}^{\infty} \frac{\cos a t - \cos b t}{t} d t

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

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 ,

\frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial u}{\partial t} = 0 u (0, t) = 0, u (4, t) = 0, u (x, o) = \frac{x}{3} (16 - x^{2})

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

8 M

3 (a)

E v a l u a t e \int_{c} \frac{\sin^{6} z}{{(z - \frac{π}{6})}^{3}} w h e r e C i s | z | = 1

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

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,

\frac{\partial u}{\partial t} = C^{2} \frac{\partial^{2} u}{\partial x^{2}}

$\dfrac {\partial u}{\partial t}= C^{2} \dfrac {\partial^{2}u}{\partial x^{2}}$ 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) = \frac{s^{2}}{(s^{2} - a^{2})^{2}}

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

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) = \frac{7 z - 2}{z (z - 2) (z + 1)} a b o u t Z = - 1

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

8 M

5 (a)

S o l v e \frac{\partial^{2} u}{\partial x^{2}} - 32 \frac{\partial u}{\partial t} = 0

$Solve\ \dfrac {\partial^{2}u}{\partial x^{2}}-32 \dfrac {\partial u}{\partial t}=0$ 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{aligned} f (x) & = x, & 0 < x < \frac{π}{2} \\ = π - x, & \frac{π}{2} < x < π \end{aligned}

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

6 M

5 (c)

E v a l u a t e \int_{- \infty}^{\infty} \frac{x^{2} d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}

$Evaluate \ \int^{\infty}_{-\infty}\dfrac {x^{2}dx}{(x^{2}+a^{2})(x^{2}+b^{2})}$ 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) = {(\frac{π - x}{2})}^{2}

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

\frac{π}{6} = \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . . . .

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

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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