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ODE602S - ORDINARY DIFFERENTIAL EQUATIONS - 1ST OPP - NOVEMBER 2023 |
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nAml BIA UnlVERSITY
OF SCIEnCEAno TECHnOLOGY
Facultyof Health,Natural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
PrivateBag13388
Windhoek
NAMIBIA
T: •264 612072913
E: msas@nust.na
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QUALIFICATION): BACHELOR of SCIENCE IN APPLIED MATHEMATICS AND STATISTICS AND
BACHELOR of SCIENCE
QUALIFICATION CODE: 07BSAM ,07BSOC
COURSE: ORDINARY DIFFERENTIAL EQUATIONS
DATE: NOVEMBER 2023
DURATION: 3 HOURS
LEVEL: 6
COURSE CODE: ODE602S
SESSION: 1
MARKS: 80
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY: EXAMINATION QUESTION PAPER
Prof Adetayo S. Eegunjobi
Prof Sunday A. Reju
INSTRUCTIONS
1. Answer any four questions on the separate answer sheet.
2. Pleasewrite neatly and legibly.
3. Do not use the left-side margin of the exam paper. This must be allowed for the examiner.
4. No books, notes and other additional aids are allowed.
5. Show all your working /calculation steps.
PERMISSIBLE MATERIALS
1. Non-Programmable Calculator
ATTACHMENTS
1. None
This paper consists of 3 pages including this front page
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ODE 602S
Ordinary Differential Equations
November 2023
1. Solve the following initial value problems:
(a) y'(x) + 1y(x) = 6x - 5, y(l) = 1, for x > 0
(5)
(b) y'(x) + y(x) tanx = e2x cosx, y(0) = 2
(5)
(c) Cobalt-60, a radioactive element employed in medical radiology, possesses a half-life
of 5.3 years. Let's consider an initial cobalt-60 sample weighing 100 grams.
i. Caculate the decay constant and derive an equation representing the quantity
of the sample that will remian t years from now.
(5)
ii. What is the time required for 85% of the sample to undergo decay?
(5)
2. (a) Find the values of o: such that y(x) = e0 x is a solution of
y"(x) - y'(x) - 6y(x) = 0.
Determine if the solutions are linearly independent or not. Hence or otherwise,
write the general solution.
(6)
(b) Given that
ay"(x) + by'(x) + cy(x) = 0
1. Write down the atndliary equation.
(2)
11. If the roots of the auxiliary equation are complex and denoted by m 1 = o: + (3i
and m2 = a - (3i, show that the general solution is
(6)
(c) Find the particular solution of the following differential equations, using undeter-
mined coefficients
y"(x) - 6y'(x) + 8y(x) = 3cosx
(6)
3. (a) Find the general solution of
0.5yiv(x) + y"(x) + 0.5y = 0
(6)
(b) Find the general solution of
2y"'(x) + 6y"(x) - Sy= 0
(c) Find the general solution of
l8x 2 y"(x) + 30xy'(x) + lOy(x) = 0, x > 0
(6)
2
(8)
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ODE 602S
Ordinary Differential Equations
November 2023
4. (a) Use Laplace Transform to solve the differential equation:
y11(t) - 4y(t) = 24cos2t, y(0) = 3, y'(0) = 4
(10)
(b) Solve by using Laplace Transform the following simultaneous differential equations:
x'(t) = x(t) - 2y(t), and y'(t) = 5x(t) - y(t), x(0) = -1, y(0) = 2
(10)
5. (a) Use Laplace transform to find the exact value of
t (xi cos 6t - cos 4t d
la
t.
(5)
(b) Find the first five terms in the series solution of
y'(x) + y(x) + x2y(x) = sinx, with y(0) = a.
(5)
(c) If J(t) = e3t and g(t) = e7t
i. Find the convolution of J(t) ® g(t)
(5)
ii. Find .C{J(t) ® g(t)}
(5)
End of Exam!
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