## Applied Mathematics 3 - May 2014

### Computer 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 $$L^{-1}\left[\frac{e^{-\pi{}s}}{s^2+2s+2}\right]$$(5 marks)
**1 (b)** State true or false with proper justification "There does not exist ar. Analytic function whose real part is x^{3} - 3x^{3}y-y^{3}.(5 marks)
**1 (c) ** prove that $$f_1\left(x\right)=1,\ f_2\left(x\right)=x,\ f_3\left(x\right)=\frac{\left(3x^2-1\right)}{2}$$ are orthogonal over (-1,1)(5 marks)
**1 (d) ** Using Green's theorem in the plane, evaluate $$
\int_c^{\ }\left(x^2-y\right)dx+\left(2y^2+x\right)\
$$ by around the boundry of the region defined by y=x^{2} and y=4.(5 marks)
**2 (a) ** Find the fourier cosine integral representation of the function f(x)=e^{-ax}, x>0 and hence show that

$$\int_0^{\infty{}}\frac{\cos{ws}}{1+w^2}dw=\frac{\pi{}}{2}e^{-x},\ x\geq{}0$$(6 marks)
**2 (b)** Verify laplace equations for $$U=\left(r+\frac{a^2}{r}\right)\cos{\theta{}}$$ Also find V and f(z).(6 marks)
**2 (c) ** Solve the following equation by using laplace transform $$\frac{dy}{dt}+2y+\int_0^tydt=\sin t$$ given that y(0)=1(8 marks)
**3 (a)** Expland $$
f\left(x\right)=\left\{\begin{array}{l}\pi{}x,\ 0<x<1 \\
0,\ 1<x<2\end{array}\right.
$$ with period 2 into a fourier series.
(6 marks)
**3 (b)** A vector field is given by $$\bar{F}=\left(x^2+xy^2\right)i+\left(y^2+x^2y\right)j\ show\ that\ \bar{F}$$ is irrotational and find its scalar potential.(6 marks)
**3 (c) ** Find the inverse z-transform of- $$f\left(Z\right)=\frac{z+2}{z^2-2z+1},\left\vert{}z\right\vert{}>1$$(8 marks)
**4 (a)** Find the constants 'a' and 'b' so that the surface ax^{2}-byz=(a+2) x will be orthogonal to the surface 4x^{2}y+z^{3}=4 at (1, -1, 2)(6 marks)
**4 (b)** $$
Given\ \ L\left(erf\sqrt{t}\right)=\frac{1}{S\sqrt{S+1}},\ evaluate\
\int_0^{\infty{}}te^{-t}erf (\sqrt{t})\ dt
$$
(6 marks)
**4 (c) ** Obtain the expansion of f(x)=x(π - x), 0<x<π as="" a="" half-range="" cosin="" series="" hence="" show="" that="" $$="" \left(i\right)\="" \sum_1^{\infty{}}\frac{{\left(-1\right)}^{n+1}}{n^2}="\frac{{\pi{}}^2}{12}" $$<br=""> $$
\left(ii\right)\ \ \sum_1^{\infty{}}\frac{1}{n^4}=\frac{{\pi{}}^4}{90}
$$</x<π><>(8 marks)
**5 (a)** If the imaginary part of the analytic function $$W=f\left(z\right)is\ V=x^2-y^2+\frac{x}{x^2+y^2}$$ find the real part U.(6 marks)
**5 (b)** If f(k)=4^{k} U(K) and g(k)=5^{k} U(K), then find the z-transform of f(k)· g(k)
(6 marks)
**5 (c) ** Use Gauss's Divergence theorem to evaluate $$\iint_s^{\ }\ \bar{N}\cdot{}\bar{F\ }ds\ \ \ where\ \bar{F}=4xi+3yj-2zk$$ and S is the surface bounded by x=0, y=0, z=0 and 2x+2y+z=4(8 marks)
**6 (a)** Obtain complex form of Fourier series for f(x)= c0s h 3x sin h 3x in (-3,3).(6 marks)
**6 (b)** Find the inverse Laplace transform of $$\frac{{\left(S-1\right)}^2}{{\left(s^2-2s+5\right)}^2}$$(6 marks)
**6 (c) ** Find the bilinear transformation under which 1, I, -1 from the z-plane are mapped onto 0,1,&infin of w-plane. Also show that under this transformation the unit circle in the w-plane is mapped onto a straight line in the z-plane. Write the name of this line.(8 marks)