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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