1
Find a particular integral of the differential equation (D

^{3}+6D+5)y=e.
2 M

2
Transform the differential equation x

^{2}y''-xy'+2y=0 with constant coefficients.
2 M

3
Find \[\nabla (\nabla . ((x^2 - yz) \bar{i} + (y^2 -xz)\bar{j} + (z^2 -xy) \bar{k})) \] at the point (1, -1, 2).

2 M

4
State Green's theorem in the plane.

2 M

5
Give an example of a complex-valued function which is differentiable at a point but not analytic at that point.

2 M

6
\[ If \ u(x,y)= 3x^2 y + 2x^2 - y^3 - 2y^2, \] verify whether u is harmonic.

2 M

7
State Cauchy's integral formula.

2 M

8
Find the residue of \[ \left \{ \dfrac {\sin 3z}{z^6} \right \} \ at \ z=0 \]

2 M

9
State sufficient conditions for the existence of Laplace transform.

2 M

10
State final value theorem.

2 M

Answer any one question from Q11 (a) & Q11 (b)

11 (a) (i)
Solve the differential equation y''+a

^{2}y=tan ax by variation of parameters method.
8 M

11 (a) (ii)
Solve the following simultaneous differential equations. \[ \dfrac {dx}{dt} + 2x -3y = t \ and \ \dfrac {dy}{dt}- 3x + 2y =e^{2t} \]

8 M

11 (b) (i)
Solve ((x+1)

^{2}D^{2}+(x+1)D+1)y=4 cos log (x+1).
8 M

11 (b) (ii)
Solve (D

^{2}-7D-6)y=(1+x)e^{2x}.
8 M

Answer any one question from Q12 (a) & Q12 (b)

12 (a) (i)
Prove that \[(\phi \bar{F})= \phi div \bar{F}+ \nabla \phi\cdot \bar{F} \] Also determine the value of n for which \[ r^{n}\bar{R}\ is \ solenoidal \ where \ \bar{R} = x\bar{i}+ y\bar{j}+ z\bar{k} \ and \ r=|\bar{R}|. \]

8 M

12 (a) (ii)
Verify Gauss divergence theorem for \[ \bar{F}= x^2 \overrightarrow{i}+ y^2 \overrightarrow{j} + z^2 \overrightarrow{k} \] over the volume of the cuboid formed by the planes x=0, x=a, y=0, y=b, z=0 and z=c.

8 M

12 (b) (i)
Prove that \[ \bar{F} = (y^2 + 2xz^2 ) \bar{i} + (2xy - z) \bar{j}+ (2x^2 z - y+2z) \bar{k} \] irrotational and hence find its scalar potential.

8 M

12 (b) (ii)
Verify Stokes' theorem for \[ \bar{F}= (x^2 +y^2) \bar{i} + 2xy \bar{j} \] where S is the rectangle in the xy-plane formed by the lines x=0, x=a, y=0 and y=b.

8 M

Answer any one question from Q13 (a) & Q13 (b)

13 (a) (i)
If f=u+iv is an analytic function, prove that \[ \left ( \dfrac {\partial ^2 } {\partial x^2 }+ \dfrac {\partial^2}{\partial y^2} \right )| f(z)|^2=4 |f'(z)|^2 \]

8 M

13 (a) (ii)
Find the bilinear transformation which maps the points ∞, 2, -1 to 1, ∞ and 0 respectively.

8 M

13 (b) (i)
Find the analytic function f=u+iv given that. u(x,y)=e

^{2x}(x sin 2y+ y cos 2 y).
8 M

13 (b) (ii)
If f=u+iv is analytic on a domain D and |f| is constant on D, prove that f must be a constant on D.

8 M

Answer any one question from Q14 (a) & Q14 (b)

14 (a) (i)
\[ If \ F(a)= \oint_c \dfrac {3z^2+7z+1}{z-a}dz \] where C:|z|=2 and |a|≠2, find F and F''(1-i).

8 M

14 (a) (ii)
Evaluate \[ \int^\infty_0 \dfrac {x^2}{(x^2+a^2) (x^2 +b^2)}dx \ \] by the method of contour integration, if a and b are positive.

8 M

14 (b) (i)
Find the Laurent's series of \[ f(z)= \dfrac {3z-2}{z(z^2-4)} \] valid in the region 2<|z+2|<4.

8 M

14 (b) (ii)
Using contour integration method show that \[ \int^{2x}_0 \dfrac {d\theta}{a+b \cos \theta}= \dfrac {2\pi}{\sqrt{a^2 -b^2}} \] if a>b>0.

8 M

Answer any one question from Q15 (a) & Q15 (b)

15 (a) (i)
Find the Laplace transform of \[ f(t) = \dfrac {\sin^2 t}{t} \]

4 M

15 (a) (i)
Find the value of \[ \int^\infty_0 te^{3t} \cos 2tdt \]

4 M

15 (a) (ii)
Solve y''+9y=cos 2t given that y(0)=1 and y(π/2)=-1, by given by the method of Laplace transform.

8 M

15 (b) (i)
Find \[ L^{-1} \left ( \log \dfrac {s^2 +1} { s (s+1)} \right ) \]

4 M

15 (b) (i)
Using convolution theorem, find y if \[ L(y)= \dfrac {s}{(s^2 +a^2)^2} \]

4 M

15 (b) (ii)
Find L(f(t)) if \[f(t) = \begin{cases}t & 0\le t\le a\\2a-t & a\le t\le 2a\end{cases}\] and \[f(t+2a)= f(t)\]

8 M

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