Introduction to Ordinary Differential Equations (MATS3107)
Mathematics and Computer Science - MCS
Semester: Resit
Level: 300
Year: 2016
RESIT EXAMINATION 2015/2016 SESSION
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE COURSE
CODE: MATS3107 COURSE TITLE: Introduction to Ode and Modelling
DURATION: 2 HRS Answer all questions.
1. a) Use the method of undetermined coefficients to find the particular solution of
y′′ + 2′ + =
-x
+ + 1
b) Solve the following differential equation
y′′ + ′ + = 0,
√
= 0, ′
√
= 1
ii) Given that y1(x) = x2 and y2(x) = x3 form a fundamental system of solution for
x
2
xy′′+ 4xy + 6y = 0 on (0, ∞), find the general solution x
2
xy′′+ 4xy′ + 6y =
expressing your result in terms of elementary functions
2. i) Let v be a continuous function of x ∈ [0, L] and consider the initial value problem
(IVP)
v′ = f(x, v), v(0) =a
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)≥ 0 g(x) ≤ ℎ
+
on an interval 0 ≤ x ≤ x
0
, then
g(x) ≤ h(x) +
dξ (1+5+6=12 marks)
3. i) Find the solution of the Euler’s equation
x
2
y′′ + 5xy′ + 4y = 0, x > 0, y(1) = 1, y′(1) = 3
ii) Given the differential equation
(2 + x2)y′′ - xy′ + 4y = 0
a) Show that x = 0 is a regular point of (2)
Solving (2) by means of a power series,
∑
"
#
$
b) Find the recurrence relation
c) The first terms in each of the two linearly independent solutions
d) The general solution (6+1+4+3+2 = 16 marks)
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