Introduction to Ordinary Differential Equations (MATS3107)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 300
Year: 2015
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) W h e n i s t he f i rs t o r d e r d i f f e r e nt i a l e q ua t i o n sa i d t o b e
(a) Separable (b) linear (c) homogenous
(ii) Solve the equation y′ =
, y(1) = 0
(iii) Find an integrating factor of the differential equation
(3xy + y
2
)dx + (x
2
+ xy)dy = 0
Hence find the general solution of (1)
(3+5+7=15) marks
2. (i) Consider the Euler’s equation
x
2
y′′ + axy′ + by = 0, a, b ∈ ℝ
a) Let x = e
t
. Show that
= x
,
= x
2
+x
b) Using the above substitution, convert (2) into a homogenous linear equation with
constant coefficients.
c) Using the method in (b) above, or otherwise, solve the initial value problem
x
2
y′′ + 5xy′ + 4y = 0, x > 0, y(1) = 1, y′(1) = 3
(ii) (a) When are two solutions of a differential equation said to be linearly independent?
(b) Consider the differential equation
y′′ +
y′-
y =
, x >0
Show that y
1
(x) = x and y
2
(x) =
are two linearly independent solutions of the associated
homogenous equation. Hence, using the method of variation of parameters, find the general
solution of the nonhomogeneous equation.
(3+1+6+1+7=18) marks
GOOD LUCK!!!!
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