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

 

ANSWER ALL QUESTIONS

 

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 e^λt 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 P_n (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 x_n(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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