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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 Transform the equation $(2x+3)^2 \dfrac {d^2y}{dx^2}-2 (2x+3)\dfrac {dy}{dx}-12 y =6x$ into a different equation with constant coefficients.
2 M

2 Find the particular integral of (D-1)2 y=ex sin x.
2 M

3 Find the ? such that $\bar{F}= (3x+2y+z)\bar{z}+ (4x+\lambda y-z)\bar{j}+ (x-y+2z)\bar{k}$ is solenoidal.
2 M

4 State Gauss divergence theorem.
2 M

5 State the basic difference between the limit of a function of a real variable and that of a complex variable.
2 M

6 Prove that a bilinear transformation has atmost two fixed points.
2 M

7 Define singular point.
2 M

8 Find the residue of the function $f(z)= \dfrac {4}{z^3 (Z-2)}$ at a simple pole.
2 M

9 State the first shifting theorem on Laplace transforms.
2 M

10 verify initial value theorem for f(t)=1+et (sin t+ cos t).
2 M

Answer any one question from Q11 (a) & Q11 (b)
11 (a) (i) Solve (D2+a2)y=sec ax using the method of variation of parameters.
8 M
11 (a) (ii) Solve: (D2-4D+3)y=ex cos 2x.
8 M
11 (b) (i) Solve the differential equation $\left ( x^2 D^2 -xD +4 \right )y=x^2 \sin (\log x)$
8 M
11 (b) (ii) Solve the simultaneous differential equations $\dfrac {dx}{dt}+2y = \sin 2t, \ \dfrac {dy}{dt}- 2x = \cos 2t.$
8 M

Answer any one question from Q12 (a) & Q12 (b)
12 (a) (i) Show that $\overrightarrow{F}= \left ( y^2 + 2xz^2 \right )\bar {i} \left (2xy -z \right )\overrightarrow{j}+ \left ( 2x^2z-y+2z \right )\overrightarrow{k}$ is irrotational and hence find its scalar potential.
8 M
12 (a) (ii) Verify Green's theorem in a plane for $\displaystyle \int_c \left [ \left (3x^2 -8y^2 \right )dx+(4y-6xy)dy \right ],$ where C is the boundary of the region defined by x=0, y=0 and x+y=1.
8 M
12 (b) (i) Using Stroke's theorem evaluate $\displaystyle \int_c \overrightarrow{F}\cdot d\overrightarrow{r} \ where \ \overrightarrow{F}= y^2 \overrightarrow{i}+ x^2 \overrightarrow{j}- (x+z) \overrightarrow{k}$ and 'C' is the boundary of the triangle with vertices at (0,0,0), (1,0,0),(1,1,0).
8 M
12 (b) (ii) Find the work done in moving a particle in the force field given by $\overrightarrow{F}=3x^2\overrightarrow{i}+ (2xz-y)\overrightarrow{j}+ z\overrightarrow{k}$ along the straight line from (0,0,0) to (2,1,3).
8 M

Answer any one question from Q13 (a) & Q13 (b)
13 (a) (i) Prove that every analytic function w=u+iv can be expressed as a function of z alone, not as a function of z
8 M
13 (a) (ii) Find the bilinear transformation which maps the points z=0,1,? into w=i,1, -1 respectively.
8 M
13 (b) (i) If f(z) is an analytic function of z, prove that $\left (\dfrac{\partial^2}{\partial x^2}+ \dfrac {\partial^2}{\partial y^2} \right )\log |f(z)|=0$
8 M
13 (b) (ii) Show that the image of the hyperbola x2-y2=1 under the transformation ω=1/z is the lemniscate r2 = cos 2θ
8 M

Answer any one question from Q14 (a) & Q14 (b)
14 (a) (i) Evaluate $\displaystyle \int_c \dfrac {zdz}{(z-1)(z-2)^2} \ where \ C \ is \ |z-2| = \dfrac {1}{2}$ by using Cauchy's integral formula.
8 M
14 (a) (ii) Evaluate $f(z)= \dfrac {1}{(z+1)(z+3)}$ in Laurent series valid for the regions |z|>3 and 1<|z|<3.
8 M
14 (b) (i) Evaluate $\int_c \dfrac {z-1}{(z+1)^2(z-2)}dz$ where C is the circle |z=i|=2 using Cauchy's residue theorem.
8 M
14 (b) (ii) Evaluate $\int^{\infty}_{0}\dfrac {\cos mx}{x^2+a^2}dx$ using contour integration.
8 M

Answer any one question from Q15 (a) & Q15 (b)
15 (a) (i) Apply convolution theorem to evaluate $L^{-1} \left [ \dfrac {s}{\left ( s^2+a^2 \right )^2} \right ]$
8 M
15 (a) (ii) Find the Laplace transform of the following triangular wave function given by $f(t)\left\{\begin{matrix}t, & 0\le t \le \pi \\ 2\pi-t, & \pi \le t \le 2\pi\end{matrix}\right. \ and \ f(t+2\pi)=f(t)$
8 M
15 (b) (i) Find the Laplace transform of $\dfrac {e^{at}-e^{-bt}}{t}$
4 M
15 (b) (ii) Evaluate $\int^{\infty}_0 te^{-2t} \cos t \ dt$ using Laplace transform.
4 M
15 (b) (iii Solve the differential equation $\dfrac {d^2 y}{dt^2}-3 \dfrac {dy}{dt}+2y=e^{-t}$ with y(0)=1 and y'(0)=0 using Laplace transform.
8 M

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