Madurai Kamraj University (MKU) 2006 M.Sc Mathematics "NUMERICAL METHODS AND DIFFERANTIAL EQUATIONS" - Question Paper
This Is for the MK DDU - "MSC Maths In in MKU", and please Refer to the Attached File,
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It's For "Msc in MATHS in MKU", It'a Course in "Madurai Kamaraj University" or %MK University%
The paper Name Is "NUMERICAL METHODS AND DIFFERANTIAL EQUATIONS"
13. (a) Piwe that a function $ is a solution of the initial Value problem y'-f(x,y),y(x0) = y0 on an interval of / if and only if it is a solution of the integral
X
equation y = y0 + J/Xf, y)dt on I.
*0
(b) Show that fix, y) = xy satisfies a Lipschitz condition on any strip a<x< b-a <y<a but not on the entire plane.
14. Describe Charpit's method for solving a first order partial differential equation. Use this method to solve p2x + q*y = z.
Paper IV NUMERICAL METHODS AND DIFFERENTIAL EQUATIONS
(For those who joined in July 2003 and after)
Time : Three hours Maximum : 100 marks
SECTION A (4 x 10 = 40 marks)
Answer any FOUR questions.
Each question carries 10 marks.
1. (a) If <j){x) is a continuous function in some interval [a, 6] that contains the root and \<f>\x)\<c <1 in this interval, then prove that for any choice of x0 6 [a, 6], the sequence {*x} determined from
xK+1 =0(xK), K =0,1,2, converges to the root of
X = (/)(x) .
(b) Explain the Netwon-Raphson method.
2. (a) Explain Gauss elimination method.
(b) State and prove Brauer theorem.
3. (a) Derive the Hermite interpolating polynomial.
(b) Write down the properties of the Chebyshev polynomial Tn(x).
4. (a) Define Lobotto Integration method and Radu Integration method.
(b) Derive the composite Simpson's rule formula.
5. (a) (i) State the Existence and Uniqueness theorem.
(ii) Define mesh points and mesh spacing.
(b) Explain Predictor-Corrector method.
6. If be n solutions of
L(y) = y(n)+a1(x)y{n~1)+ --- + an(x)y = 0 on an interval I and if x0 be any point in I, then prove that
W(*\, ,(*>)(*) = exp
x
- ax(.t)dt
W(,-.4,)(*<))
,_3x - 2xy .
7. Verify whether y'=
is an exact equation
x -2y
and then solve it.
8. Find the general integral of the linear partial
2 2
differential equation z(xp -yq) = y -x
SECTION B (3 x 20 = 60 marks) , Answer any THREE questions. t Each question carries 20 marks.
9. (a) Find the root of the equation cosx-xex =0 using the Secant and Regula-Falsi method.
(b) Find the largest eigen-value in modulus and the corresponding eigenvector of the matrix.
-15 4 3 10 -12 6 20 -4 2
using the power
A =
method.
10. Evaluate the integral I = - using
J 1 + x
o
(a) composite trapezoidal rule (b) composite Simpson's rule, with 2, 4 and 8 equal subintervals.
3 6559/KA4
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