Introduction to Ordinary Differential Equations (MATS3107)

Mathematics and Computer Science - MCS

Semester: First Semester

Level: 300

Year: 2016

Answers (200 )
UNIVERSITY OF BAMENDA
FACULTY OF SCIENCE
FIRST SEMESTER EXAMINATION
DEPARTMENT: MATHEMATICS COURSE INSTRUCTOR: Tanyu nee Nfor Vivian
MONTH:
March
COURSE CODE & NUMBER :
MATS3107
YEAR:
2016
COURSE TITLE:
Introduction to ODE and Modelling
DATE:
05/02/2016
CREDIT VALUE:
Six
TI M E ALLO WED:
3 hour
I N S T R U C T I O N : A n s w e r a l l t h e q u e s t i o n s . Al l n e c e s s a r y w o r k m u st b e s h o w n a n d m u s t
b e ne a t l y a n d o r d e r ly p r e s e n t e d .
1 . ( i ) L e t v b e a c o n ti n u o u s fu n c t i o n o f x [0 , L ] an d c o n s id e r t h e i n i t i al v a lu e
( IV P )
v ′ = f (x , v ), v ( 0 )
a) Define an integral solution of (1)
b) Suppose f is continuous. Show that the IVP (1) is equivalent to the integral equation
(1)
ii) Show that if real-valued continuous function f(x), g(x) and h(x) satisfy the inequalities
f(x) g(x)
+

on an interval 0x x
0
, then
g(x) h(x) +

dξ
iii) Assume that the rate at which radioactive nuclei decay is propositional to the number of such
nuclei that are present in the given sample. In a certain sample 10 percentage of the original number
of radioactive nuclei have undergone disintegration in a period of 100 years
a) What percentage of the original radioactive nuclei will remain after 100 years?
b) In how many years will only one-fourth of the original number remain? (1+5+6+3+2=17
marks)
2. (i) a) Define a contraction A of a metric space (X, d)
b) Let A be contraction on a complete metric space (X, d), prove that A has a unique fixed
point.
(ii) Consider the differential equation
y′ + p(x)y+ q(x)y = 0
where p and q are continuous on an opera interval I. Let y
1
and y
2
be two solutions of (2). Prove
that the wronkian W [y
1
, y
2
] is either zero or never zero
(iii) Use Picard’s method of successive approximation to find the general solution of the IVP


= 2x(y + 1), y(0) = 0
(1+5+5+5=16 marks)
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3. (i) Show that
2x
+ (1 + x)


+ y = 0 has a regular point at 0 and determine
a) The indicial equation
b) The roots of the indicial equation
c) The recurrence relation
d) Write down a series solution of the equation (not necessary the general equation
(ii) Suppose M : D , (x,y) M(x,y), N : D , (x,y) N(x,y) , D 
2
and M and N
are continuous on D. Prove that M(x,y)dx + N(x,y)dy is exact

=
!

(2+2+2+2+5 = 13 marks)
4. (i) When is a first order differential equation y′ = f(x, y) said to be
(a) Separable (b) linear (c) homogenous
(ii) Classify according to the classes in (i), and hence solve each of the following
(a) y′ =
"#$
#"
(b)


=
#
%
(c)


(1 + e
x
cos y) + e
x
sin y -2x =0
(4 + 16 = 20 marks)
Good Luck
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