LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – November 2008
MT 1806 – ORDINARY DIFFERENTIAL EQUATIONS
Date : 081108 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
ANSWER ALL QUESTIONS
 a) (i) If the Wronskian of 2 functions x_{1} and x_{2} on I is nonzero for at
least one point of the interval I, show that x_{1} and x_{2} are linearly
independent on I. Hence show that sin x, sin 2x, sin 3x are
linearly independent on [ 0, 2 ].
OR
(ii) Suppose x_{1} (t) and x_{2} (t) satisfy a x”(t) + b x'(t) + c x(t) = 0,
where a is not zero, Show that A x_{1} (t) + B x_{2} (t) satisfies the
differential equation. Verify the same in x” + λ^{2} x = 0. (5 Marks)
 b) (i) State and prove the Abel’s Formulae. (8 Marks)
(ii) Solve x” – x’ – 2x = 4t^{2} using the method of variation of
parameters. (7 Marks)
OR
(iii) If λ is a root of the quadratic equation a λ^{2} + b λ + c = 0,
prove that e^{λt} is a solution of a y” + by’ + c y = 0. (15 Marks)
 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)
 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)
 b) (i) State and prove Rodriguez’s Formula and find the value of
{8 P_{4} (x) + 20 P_{2} (x) + 7 P_{0} (x)}
OR
(ii) Show that P_{n }(x) = F_{1 }[n, n+1; 1; (1x)/2] (15 Marks)
 a) (i) Considering the differential equation of the SturmLiouville
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)
 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)
 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 ≥ t_{0} . (5 marks)
 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)
Latest Govt Job & Exam Updates: