Tamil Nadu Open University (TNOU) 2009-2nd Year M.Sc Mathematics Tamilnadu open university Maths Differential equation - Question Paper

M.Sc. DEGREE EXAMINATION
JUNE 2009.

Second Year

(AY 200607 batch onwards)

Mathematics

DIFFERENTIAL EQUATIONS

Time : 3 hours Maximum marks : 75

PART A (5 5 = 25 marks)

Answer any FIVE questions.

1.         Discuss the solution of the differential equation where , are constants.

2.         Solve the equation .

3.         If are Legendre polynomials, prove that .

4.         Determine the regular singular point of the equation

             .

5.         Let be a fundamental matrix of the system of equations and let be a constant
non-singular matrix. Prove that is also a fundamental matrix of above system.

6.         Find a fundamental matrix of the equation

7.         Solve the equation .

8.         Derive the elementary solution of the Laplace equation .

PART B (5 10 = 50 marks)

Answer any FIVE questions.

9.         Let be any solution of on an interval containing a point . Then prove that for all in ,

             where .

10.       Find the solution of the initial value problem , , .

11.       Derive the power series solution of the equation , where is a constant.

12.       Determine the solution of Bessel equation when .

13.       State and prove the existence and uniqueness theorem for the IVP , .

14.       Let be periodic with period . Let be an constant matrix. Prove that a solution of is periodic of period if and only if .

15.       Reduce the equation

             into its canonical form.

16.       State and prove Kelvins inversion theorem.