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1
Find the particular integral of (D2-2D+1)y=cosh x
2 M
2
\[ Solve \ x^2\dfrac {d^2y}{dx^2}+4x \dfrac {dy}{dx}+2y=0 \]
2 M
3
Find the directional derivative of ?=xyz at (1,1,) in the direction of\[ \overrightarrow {i}+ \overrightarrow{j}+ \overrightarrow{k} \]
2 M
4
If \[ \overrightarrow{A} \ and \ \overrightarrow{B}\] are irrotational, prove that \[ \overrightarrow{A}\times \overrightarrow{B} \] is solenoidal.
2 M
5
Find the image of the line x=k under the transformation w=1/z
2 M
6
Find the fixed points of mapping \[ w=\dfrac {6z-9}{z} \]
2 M
7
Evaluate \[ \int_c \dfrac {3z^2 + 7z+1}{z+1}dz, \ where \ C \ is |z|=\dfrac {1}{2} \]
2 M
8
Find the residue of \[ \dfrac {1-e^{2z}}{z^4} \ at \ z=0 \]
2 M
9
Find the Laplace transform of \[ \dfrac {t}{e^t} \]
2 M
10
Verify initial value theorem for the function f(t)=ae-bt
2 M
Answer any one question from Q11 (a) & Q11 (b)
11 (a) (i)
Solve the differential equation \[ \dfrac {d^2y}{dx^2}+2\dfrac {dy}{dx}+ y=\dfrac {e^{-x}}{x^2} \] by the method of variation of parameters.
8 M
11 (a) (ii)
\[ Solve : \ (3x+2)^2 \dfrac {d^2y}{dx^2}+3 (3x+2)\dfrac {dy}{dx} -36y = 3x^2 +4x+1 \]
8 M
11 (b) (i)
Solve the simultaneous differential equations: \[ \dfrac {dx}{dt}+5x-2y=t; \ \dfrac {dy}{dt}+2x+y=0 \]
8 M
11 (b) (ii)
\[ Solve \ x^2 \dfrac {d^2 y}{dx^2}+ 4x \dfrac {dy}{dx}+2y=x^2 + \dfrac {1}{x^2} \]
8 M
Answer any one question from Q12 (a) & Q12 (b)
12 (a)
Verify Stokes' Theorem for the vector field \[ \overrightarrow{F}= (2x-y)\overrightarrow {i}-yz^2 \overrightarrow{j}-y^2z\overrightarrow{k} \] over the upper half surface x2+y2+z2-1, bounded by its projection on the xy-plane.
16 M
12 (b)
Verify divergence theorem for \[ \overrightarrow{F}=x^2 \overrightarrow{i}+z\overrightarrow{j} + yz\overrightarrow{k} \] over the cube formed by the plane x=±1, y=±1, z=±1.
16 M
Answer any one question from Q13 (a) & Q13 (b)
13 (a) (i)
Prove that the function u-ex(x cos y -y sin y) satisfies Laplace's equation and find the corresponding analytic function f(z)=u+iv.
8 M
13 (a) (ii)
Find the Bilinear transformation which maps z=0, z=1, z=∞ into the points w=i, w=1, w=-i.
8 M
13 (b) (i)
Find the image of |z-2i|=2 under the transformation w=1/z.
8 M
13 (b) (ii)
If f(z) is an analytic function of z, prove that \[ \left ( \dfrac {\partial^2}{\partial x^2}+ \dfrac {\partial^2}{\partial y^2} \right )|f(z)|^2=4 |f(z)|^2 \]
8 M
Answer any one question from Q14 (a) & Q14 (b)
14 (a) (i)
Expand the function \[ f(z)= \dfrac {z^2-1}{z^2+5z+6} \] in Laurent's series for |z|>3
8 M
14 (a) (ii)
Evaluate \[ \int_c \dfrac {\sin \pi z^2+ \cos \pi z^2 }{(z+1)(z+2)}dz \] where C is |z|=3
8 M
14 (b) (i)
Evaluate \[ \int^{\infty}_0 \dfrac {x^2 dx}{(x^2 +a^2)(x^2+b^2)}, \ a>0,b>0 \]
8 M
14 (b) (ii)
Evaluate \[ \int^{2\pi}_0 \dfrac {\cos 3\theta}{5-4\cos \theta}d\theta \] using contour integration.
8 M
Answer any one question from Q15 (a) & Q15 (b)
15 (a) (i)
\[ Find \ L\left [ t^2e^{-3t} \sin 2t \right] \]
8 M
15 (a) (ii)
Find the Laplace transform of the square-wave function (or Meoander function) of period α defined as \[ f(t)= \left\{\begin{matrix}1 & when & 0<t<\frac{a}{2} \\-1, &when & \frac {a}{2} <t<a\end{matrix}\right. \]
8 M
15 (b) (i)
Using convolution theorem find the inverse Laplace transform of \[ \dfrac {4}{(s^2+2s+5)^2} \]
8 M
15 (b) (ii)
Solve y''+5y'+6y=2 given y'(0)=0 and y(0)=0 using Laplace transform.
8 M
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