Kerala University 2005 B.Sc Mathematics differential equations------------------- - Question Paper
k
R* N..................................... ......................(2 pages) K 5104
Name........................................................
FINAL YEAR B.Sc. DEGREE EXAMINATION, MARCH/APRIL 2005
Part IIIGroup IMathematics Paper IVDIFFERENTIAL EQUATIONS, NUMERICAL ANALYSIS AND VECTORS Time : Three Hours Maximum : 65 Marks
A maximum of 13 marks can be earned from each unit.
Unit I
3. Show that the differential equation (x2-4xy + Zy1) dx + (y2 + 4xy -2 x2) dy = 0 is exact, and hence solve it. '
(4 marks) (3 marks)
4. Solve (D2 - 2 D + l)y - e31, where D = .
fS-m&rka)1
5. Solve
y cos x .
Unit II
6. Solve x2 + x - 9y = log*. dx2 dx
7. Solve = * + 53f ; - 3 y.
at at
8. Find the Laplace transform of t sin t.
9. Find :
(a) L
-i 2s
(s-l)<*-2)2
(4 marks)
(6 marks) (4 marks)
(b) L
(3 + 3 = 6 marks) Turn over
2 K 5104
Unit m
10. Define the shift operator E and central difference operator 5 show that 8 = EV2 - E-V2.
(5 marks)
11. Derive Gauss backward formula for interpolation. (5 marks)
12. Find x at y = 1.4 from the following data :
Y 1.2 2.0 2.5 3.0
X : 1.34 0.57 0.33 0.21
(4 marks)
13. Show that the n* divided differences of a polynomial of 71th degree are constant. (6 marks)
Unit IV
14. Show that [o + b, b + c, c + a] = 2 ja b c]. (4 marks)
15. If F = (* + y + 1) i + j - (x + y) k , show that p _ cur| _ q (4 marks)
16. Determine the constant o so that the vector f = (x + 3y) i + (y - 2z)j + (x + az) k is solenoidal.
(4 marks)
17. Ifi|>(x, y,z) = 3x?y-y*z2, find grad i)> at(l, -2, -1). (5 marks)
18. If a, b, c and are coplanar vectors, show that (a x ft) x (c x dj = 0. (3 marks)
Unit V
19. Evaluate Jr . dr where C is the helical path x = cos t,y = sin t,z = t joining the points determined
c
by t = 0 and t = tc/4 .
(5 marks)
20. Find 01* i + xj + y ft]. n ds where S is the quadrant of the circle x? + y2 = 1 between the
s
positive parts of the axes.
(5 marks)
21. State Gauss Divergence theorem. (3 marks)
22. State Stoke's theorem. Verify the theorem for the function F = (2* - y) i - yz2) - y2zk where
S is the upper half of the sphere x2 + y2 + za = 1 and C is its boundary.
(2 + 5 = 7 marks)
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