STUPIDSID
1
Solve (D2-4)y=1
2 M
2
Convert (3x2D2+5xD+7)y=2/x log x into an equation with constant coefficients.
2 M
3
Define solenoidal vector function. \[ If \ \overrightarrow {V}= (x+3y)\overrightarrow{i}+ (y-2z)\overrightarrow{j}+ (x+2\lambda z)\overrightarrow{k} \] is solenoidal, find the value of ?
2 M
4
State Green's theorem.
2 M
5
Find the constants a,b if f(z)=x+2ay+i(3x+by) is analytic.
2 M
6
Find the critical points of the transformation \[ w=1+\dfrac {2}{z} \]
2 M
7
Evaluate \[ \int_c \dfrac {z+4}{z^2+2z} \] where C is the circle \[ \left |z-\dfrac {1}{2} \right |=\dfrac {1}{3} \]
2 M
8
Find the residue of \[ f(z)= \dfrac {1-e^{-z}}{z^3} \ at \ z=0 \]
2 M
9
Find the Laplace transform of \[ f(t)=\dfrac {1-e^{-t}}{t} \]
2 M
10
Find the Laplace transform of the function \[ f(t)\left\{\begin{matrix} 1,&t=0 \\0, &t \ne 0 \end{matrix}\right. \]
2 M
11 (a) (ii
Solve by the method of variation of parameters
\[ 2 \dfrac {d^2y}{dx^2}+ 8y=\tan 2x \]
\[ 2 \dfrac {d^2y}{dx^2}+ 8y=\tan 2x \]
8 M
Answer any one question from Q11 (a) & Q11 (b)
11 (a) (i)
\[ Solve \ \dfrac {d^2y}{dx^2}- 2 \dfrac {dy}{dx}+ y=8xe^x \sin x \]
8 M
11 (b) (i)
\[ Solve \ x^2 \dfrac {d^2y}{dx^2}+ 4x\dfrac {dy}{dx}+ y=e^{\log x} \]
8 M
11 (b) (ii)
\[ Solve \ \dfrac {dx}{dt}+4x+3y=t; \ \dfrac {dy}{dt}+2x+5y=e^{2t} \]
8 M
Answer any one question from Q12 (a) & Q12 (b)
12 (a) (i)
Show that the vector field \[ \bar {F}= (x^2+xy^2)\bar{i}+ (y^2 + x^2y)\bar{j} \] is irrotational. Find its scalar potential.
6 M
12 (a) (ii)
Verify Stokes' Theorem for \[ \bar{F}= (x^2+y^2)\bar{i}-2xy\bar{j} \] taken around the rectangle formed by the line x=-a; x=+a, y=0 and y=b.
10 M
12 (b) (i)
Find a and b so that the surface ax3-by2z-(a+3)x2=0 and 4x2y-z3-11=0 cut orthogonally at the point (2, -1, -3)
6 M
12 (b) (ii)
Verify Gauss Divergence theorem for \[ \bar{F}=4xz\bar{i}-y^2\bar{j}+yz\bar{k} \] where S is the surface of the cube formed by the planes x=0, x=1, y=0, y=1, z=0 and z=1.
10 M
Answer any one question from Q13 (a) & Q13 (b)
13 (a) (i)
Prove that u=e-2xy sin (x2-y2) is harmonic. Find the corresponding analytic function and the imaginary part.
8 M
13 (a) (ii)
Find the bilinear map which maps the points z=0, -1, i onto the points w=i, 0, ∞. Also find the image of the unit circle of the z plane.
8 M
13 (b) (i)
Prove that \[ w=\dfrac {z}{1-z} \] maps the upper half of the z-plane to the upper half of the w-plane and also find the image of the unit circle of the z plane.
8 M
13 (b) (ii)
Find the analytic function f(z)=u+iv where v=3r2 sin 2θ-2r sin θ. Verify that u is a harmonic function.
8 M
Answer any one question from Q14 (a) & Q14 (b)
14 (a) (i)
Find the residue of \[ f(z)= \dfrac {z^2}{(z+2)(z-1)^2} \] at its isolated singularities using Laurentz's series expansion.
8 M
14 (a) (ii)
Evaluate \[ \int^{2\pi}_0 \dfrac{\cos 2 \theta} {5+4 \cos \theta}d \theta \] using contour integration
8 M
14 (b) (i)
show that \[ \int^{\infty}_{-\infty}\dfrac {x^2-x+2}{x^4+10x^2+9}dx = \dfrac {5\pi}{2} \]
8 M
14 (b) (ii)
Evaluate \[ \int_c \dfrac {z+1}{(z^2+2z+4)^2}dz \] where C is the circle |z+1+i|=2, by Cauchy's integral formula.
8 M
Answer any one question from Q15 (a) & Q15 (b)
15 (a) (i)
Evaluate \[ L^{-1} \left ( \dfrac {3s^2+16s+26}{s(s^2+4s+13)} \right ) \]
8 M
15 (a) (ii)
Find the inverse Laplace transform of the following: \[ \log \left ( \dfrac {s+1}{s-1} \right ) \]
8 M
15 (b) (i)
\[ Find \ L^{-1} \left [ \dfrac {s}{(s^2+a^2)^2} \right ] \ and \ find \ L^{-1} \left [ \dfrac {1}{(s^2+a^2)^2} \right ] \\ and \ find \ L^{-1} \left (\dfrac {1}{(s^2+9s+13)^2} \right ) \]
8 M
15 (b) (ii)
Using Laplace transform, solve y''+y'=t2+2t, y(0)=4 and y(0)=-2
8 M
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