Karnataka State Open University (KSOU) 2011-1st Sem M.Sc Mathematics (Differential Equations) - Question Paper
Illllllllllllllllllllllll Math 1.5
1 Semester M.Sc. Examination, May 2011 MATHEMATICS Differential Equations
Time : 3 Hours Max. Marks : 80
Note : IA Answer any five questions.
2 A All questions carry equal marks.
1. a) State and prove Picards theorem for existence and uniqueness of an
initial value problem. 12
b) Find the general solution of y(S) - y(4) - y' + y = 0 4
2. a) Find the general solution of y" + y = tan x sec x by the method of variation
of parameters. 8
b) Find the solution of x2 y" + 7xy' + 8y = 0 by finding the solution of its
adjoint equation by stating appropriate result. 8
3. a) Define Strum-Limville Problem. Find the eigen values and eigen function
of Strum-Limville Problem. 8
b) Solve the boundary value problem y" + Xy = f(x), y(0) = 0, y(7t) = 0 by Greens function method. 8
4. a) Find the series solution of laguerre differential equation
xy" + (1 - x)y' + ny = 0 around the singular point for a non-negative positive integer n. 9
wf-x [lifm=n
b) Prove that [ e Lm (x) Ln (x) dx = I 7
I m n Oifmn
5. a) Find the general solution of the non-homogeneous linear system
|
"-3 1" |
3t" | ||
|
= |
2 4 |
X(t) + |
-t e |
b) Define the four types of critical points of the linear autonomous system
dx
dt
dy
dt
= ax + by
= cx + dy, when ad - bc 0.
c) Determine the nature and stability of critical point of the system
dy
dx
dy 2 2
6. a) Find the critical points of the non-linear system = I - y : = x2 - y2.
Determine their nature and stability. 8
b) State only the Liapunov stability theorem. Determine the stability of the critical
dx 5 3 dy 3 3 point (0, 0) of the system = - x5 - y3 ; = 3x3 - y3. 8
7. a) State Cauchy problem and discuss its geometrical significance. 8
b) Define semilinear, quasilinear and nonlinear partial differential equations of first order. Explain the method of characteristics for solving quasilinear cauchy problem. 8
dx
8. Solve :
d3z
d3z
a3z
dx3
a)
+
- -
3 = e2x + y + sin (x+2y).
d3z
b) Discuss the method of characteristics for the cauchy problem of second order and illustrate with a suitable example. 10
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Attachment: |
| Earning: Approval pending. |
