LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 20

FIRST SEMESTER – NOV 2006

         MT 1806 – ORDINARY DIFFERENTIAL EQUATIONS

 

 

Date & Time : 31-10-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

 

 

 

 

ANSWER ALL QUESTIONS

 

1. (a) 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.

OR

Consider the Differential Equation x” + λ² x = 0, prove that

A cos λx + B sin λx is also a solution of the Differential equation.

(5 Marks)

(b) State and prove the method of variation of parameters.

OR

By the method of variation of parameters solve x”’ − x’ = t.   (15 Marks)

 

1. (a) Obtain the indicial form of the equation

2x² (d²y/dx² ) +  (dy/dx)   + y = 0

OR

Obtain the indicial form of the Bessel’s differential equation. (5 Marks)

(b) Solve the differential equation using Frobenius Method ,

x² (d²y/dx²)  + x q(x) (dy/dx)  + r(x) y = 0 and discuss about their

solutions when it’s  roots differ by an integer .

OR

Solve the  Legendre’s equation,

(1 – x²) (d²y/dx²)  – 2x (dy/dx)   + L(L+1)y = 0.                     (15 Marks)

 

 

III. (a) Prove that ∫⁺¹_-1 P_n(x) dx = 2 if n = 0 and

∫⁺¹_-1 P_n(x) dx = 0 if n ≥ 1

OR

Show that Hypergeometric function does not change if the parameter α and

β are interchanged, keeping γ fixed.                                                 (5 Marks)

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

OR

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

 

 

 

 

 

 

 

 

 

IV.(a) Considering an Initial Value Problem x’ =  -x, x(0) = 1, t ≥ 0, find x_n(t).

OR

Find the eigen value and eigen function of x” + λ x = 0, 0 < t ≤   (5 Marks)

(b) State and prove Picard’s Boundary Value Problem.

OR

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

only if  x(t) = ∫^b_a G(t,s) f(s) ds.                                                      (15 Marks)

 

V.(a) Discuss the fundamental Theorem on the stability of the equilibrium of

the system x’ = f(t, x).

OR

Obtain the condition for the null solution of the system x’ = A(t) x is

asymptotically stable.                                                                     (5 Marks)

(b) Study the stability of a linear system by Lyapunov’s direct method.

OR

Study the stability of a non-linear system by Lyapunov’s direct method.

(15 Marks)

 

 

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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 e^λt 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 P_n(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 λ_m and λ_n corresponding to eigen functions

x_m(t) and x_n(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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