# Loyola College M.Sc. Mathematics Nov 2008 Ordinary Differential Equations Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 AB 28

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

# MT 1806 – ORDINARY DIFFERENTIAL EQUATIONS

Date : 08-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

1. a) (i) If the Wronskian of 2 functions x1 and x2 on I is non-zero for at

least one  point of the interval I, show that x1 and x2 are linearly

independent on I. Hence show that sin x, sin 2x, sin 3x are

linearly independent on [ 0, 2 ].

OR

(ii) Suppose x1 (t) and x2 (t) satisfy a x”(t) + b x'(t) + c x(t) = 0,

where a is not zero, Show that A x1 (t) + B x2 (t)  satisfies the

differential equation. Verify the same in x” + λ2 x = 0.          (5 Marks)

1. b) (i) State and prove the Abel’s Formulae.                                     (8 Marks)

(ii) Solve x” – x’ – 2x = 4t2 using the method of variation of

parameters.                                                                              (7 Marks)

OR

(iii) If λ is a root of the quadratic equation a λ2 + b λ + c = 0,

prove that eλt is a solution of a y” + by’ + c y = 0.                 (15 Marks)

1. a) (i) Whenever n is a positive or negative integer,

show that .

OR

(ii) Obtain the linearly independent solution of the Legendre’s

differential equation.                                                             (5 Marks)

1. b) (i) For the differential equation

Obtain the indicial equation by the method of Frobenius. (8 Marks)

(ii) Prove that                                (7 Marks)

OR

(iii) Solve the Bessel’s equation  .      (15 Marks)

III. a) (i) Express x4 using Legendre’s polynomial.

OR

(ii) Show that F ( 1; p; p; x ) = 1/ (1 – x )                                        (5 Marks)

1. b) (i) State and prove Rodriguez’s Formula and find the value of

{8 P4 (x) + 20 P2 (x) + 7 P0 (x)}

OR

(ii) Show that Pn (x) =  F1 [-n, n+1; 1; (1-x)/2]                             (15 Marks)

1. a) (i) Considering the differential equation of the Sturm-Liouville

problem, prove that all the eigen values are real.

OR

(ii) Considering an Initial Value Problem x’ =  2x, x(0) = 1, t ≥ 0, find xn(t).

(5 Marks)

1. b) (i) State Green’s Function. Show that x(t) is a solution of L(x) + f(t) = 0 if

and only if  .

OR

(ii) State and prove Picard’s initial value problem.                      (15 Marks)

1. a) (i) Write down Lyapunov’s stability statements.

OR

(ii) Prove that the null solution of x’ = A (t) x is stable if and only if there

exists a positive constant k such that    | φ | ≤ k, t ≥ t0 .            (5 marks)

1. b) (i) State and prove the Fundamental Theorem on the stability of the

equilibrium of a system x’ = f (t, x).

OR

(ii) Discuss the stability of a linear system x’ = A x  by

Lyapunov’s Direct Method.

(15 Marks)

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