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

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 27

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2008

    MT 1806 – ORDINARY DIFFERENTIAL EQUATIONS

 

 

 

Date : 03/05/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

ANSWER ALL QUESTIONS

 

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

where ‘a’ is not zero then show that A x1 (t) + B x2 (t)  satisfy the Differential Equation.

 

OR

If the Wronskian of 2 functions x1(t) and x2(t) on I is non-zero for at least one point of the interval

I, show that x1(t) and x2(t) are linearly independent on I.                                                       (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 x” + bx’ + c x = 0.                 (15 Marks)

 

  1. a) Prove that exp[ x/2( t – t – 1 )] = .

OR

Prove that (1 – 2tx + t2 ) – ½ = if │t│< 1 & │x│≤ 1.      (5 Marks)

 

  1. b) Solve the Legendre’s Equation ( 1 – x2 ) y´´– 2xy´ + ny = 0

OR

Solve x ( 1 – x ) y´´ + ( 1 – x ) y´ –  y =0                                        (15Marks)

 

III. (a) Prove that  = 2 if n = 0 and

= 0 if n ≥ 1

OR

Find (d/dx) F (α; β; γ; x )                                                               (5 Marks)

 

 

 

 

 

(b) Obtain Rodrigue’s Formula and hence find P0(x), P1(x), P2(x) & P3(x).

OR

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

 

 

  1. a) Considering the Differential Equation of the Sturm-Liouville,

show that the eigen values λm and λn corresponding to eigen functions

xm(t) and xn(t) are orthogonal with respect to  weight function r(t).

OR

Solve the initial value problem x´ = t + x, x(0) = 1                     (5 Marks)

 

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

if and only if x(t) = .

OR

State and prove Picard’s Initial value Problem.                         (15 Marks)

 

  1. a) Define Lyapunov’s Stability Statements.

OR

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) State and prove the Fundamental Theorem on the stability of the

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

OR

Explain the stability of Quasi-linear system x’ = A(t) x.               (15 Marks)

 

 

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