Bhavnagar University 2007 M.Sc Mathematics DIFFERENTIAL EQUATIONS AND NUMERICAL METHODS - Question Paper

M.Sc. DEGREE EXAMINATION, DECEMBER 2007.
Mathematics
DIFFERENTIAL EQUATIONS AND
NUMERICAL METHODS
Time : 3 hours Maximum : 100 marks
ans any 5 ques..
All ques. carry equal marks.
1. (a) For any real , and constants there exists a solution of of the initial value issue , on .
(b) Find the general solution of the differential formula .
2. (a) If and are possitive constants then obtain the general solution of .
(b) Solve the differential formula :
using the method of evaluation of parameters.
3. (a) Integrate the equations for the characteristics of the formula expressing and in terms of , and then obtain the solution of this formula which decrease to when .
(b) Show how to solve by Jacobi’s method, a partial differential formula of the kind.

and illustrate the method by finding a complete integral of the formula :
.
4. (a) By separating the variables, show that the
one-dimensional wave formula :
has solutions of the form exp where and are constants.
(b) Explain Boundary value issues.
5. (a) Write the subsequent algorithm. Steepest descent and Line searchly quadratic inter polation.
(b) Solve the system by fixed point iteration.
6. (a) Verify directly that the Legendre polynomial is orthogonal to any polynomial of degree 2.
(b) Explain Least-squares approximation by polynomials.
7. (a) Explain Gaussian Rules in Numerical Integration.
(b) Find the general solution of the equations :
.
8. (a) By dividing the range into ten equal parts evaluate by Trapezoidal and Simpson’s rule. Verify your ans with integration.
(b) Derive the rectangle rule.

Earning:   Approval pending.