Applied Mathematics 3 - May 2016

MU Information Technology (Semester 3)

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) If ∫∞0e−2tsin(t+α)cos(t−α)dt=14,find α∫0∞e−2tsin⁡(t+α)cos⁡(t−α)dt=14,find α \int ^{\infty}_0e^{-2t}\sin(t+\alpha)\cos(t-\alpha)dt=\dfrac{1}{4},\text{find}\ \alpha 5 marks

1(b) Find half range Fourier cosine series for f(x) = x, 0 < x < 2 5 marks

1(c) If u(x,y) is a harmonic function then prove that f(z)=u_x - iu_y is an analytic function. 5 marks

1(d) prove that ∇f(r)=f′(r)¯rr∇f(r)=f′(r)r¯r \nabla f(r)=f'(r)\dfrac{\bar{r}}{r} 5 marks

2(a) If v = e^x siny, prove that v is a harmonic function. Also find the corresponding analytic function. 5 marks

2(b) Find Z-transform of f(k) = b^k, k≥0 5 marks

2(c) Obtain Fourier series for f(x)=3x2−6xπ+2π212 in(0,2π),f(x)=3x2−6xπ+2π212 in(0,2π), f(x)=\dfrac{3x^2-6x \pi+ 2\pi^2}{12}\ \text{in}(0,2\pi),
where f(x+2π)=f(x), hence deduce that π26=112+122+132+⋯π26=112+122+132+⋯ \dfrac{\pi^2}{6}=\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots 5 marks

3(a) Find inverse Laplace of (s+3)2(s2+6s+5)2(s+3)2(s2+6s+5)2 \dfrac{(s+3)^2}{(s^2+6s+5)^2} using Convolution theorem 5 marks

3(b) Show that the set of functions { sinx, sin3x, sin5x,....} is orthogonal over [0, π/2]. Hence construct orthonormal set of functions. 5 marks

3(c) Verify Green theorem for ∫c1ydx+1xdy∫c1ydx+1xdy \int _c \dfrac{1}{y}dx+\dfrac{1}{x}dy where C is the boundary of region defined by x=1, x=4, y=1 and y=√xy=x y=\sqrt{x} 5 marks

4(a) Find Z{ k² a^k-1 U(k-1)} 5 marks

4(b) Show that the map of the real axis of the z-plane is a circle under the transformation w=2z+iw=2z+i w=\dfrac{2}{z+i} . Find its centre and the radius. 5 marks

4(c) Express the function f(x)={sinx|x|<πo|x|>πf(x)={sin⁡x|x|<πo|x|>π f(x)=\left{\begin{matrix} \sin x & |x|<\pi\\ o & |x|>\pi \end{matrix}\right. as Fourier sine Integral. 5 marks

5(a) Using Gauss Divergence theorem evaluate ∬s¯N.¯Fds where ¯F=x2i+zj+yzk∬sN¯.F¯ds where F¯=x2i+zj+yzk \iint _s \bar{N}.\bar{F}ds \ \text{where}\ \bar{F}=x^2i+zj+yzk and S is the cube bounded by x=0, x=1, y=0, y=1, z=0, z=1 5 marks

5(b) Find inverse Z-transform of F(z)=z(z−1)(z−2), |z|>2F(z)=z(z−1)(z−2), |z|>2 F(z)=\dfrac{z}{(z-1)(z-2)}, \ |z|>2 5 marks

5(c) Solve (D²+3D+2)y = e^-2t sint, with y(0)=0 and y'(0)=0 5 marks

6(a) Find Fourier expansion of f(x) = 4-x² in the interval (0, 2) 5 marks

6(b) A vector field is given by ¯F=(x2+xy2)i+(y2+x2y)j.F¯=(x2+xy2)i+(y2+x2y)j. \bar{F}=(x^2+xy^2)i+(y^2+x^2y)j. . Show that ¯FF¯ \bar{F} is irrotational and find its scalar potential. 5 marks

6(c)(i) L−1{tan−1(as)}L−1{tan−1⁡(as)} L^{-1}\left { \tan^{-1}\left ( \dfrac{a}{s} \right ) \right } 5 marks

6(c)(ii) L−1(e−πss2−2s+2)L−1(e−πss2−2s+2) L^{-1}\left ( \dfrac{e^{-\pi s}}{s^2-2s+2} \right ) 5 marks