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ODE602S - ORDINARY DIFFERENTIAL EQUATIONS - 2ND OPP - JANUARY 2024 |
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nAml BIA UnlVERSITY
OF SCIEnCE
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics.
StatisticsandActuarialScience
13JacksonKaujeuaStreet
Private Bag 13388
Windhoek
NAMIBIA
T: •264 612072913
E: msas@nust.na
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QUALIFICATIONS: BACHELOR of SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
AND BACHELOR of SCIENCE
QUALIFICATION CODE: 07BSAM ,07BSOC
LEVEL: 6
COURSE: ORDINARY DIFFERENTIAL EQUATIONS
COURSE CODE: ODE602S
DATE: JANUARY 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
SECOND OPPORTUNITY/SUPPLEMENTARY: EXAMINATION QUESTION PAPER
EXAMINER:
MODERATOR:
Prof Adetoyo S. Eegunjobi
Prof SundayA. Reju
INSTRUCTIONS
1. Answer ALL 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
January 2024
l. (a) i. Solve the following initial value problem:
y'()X = y(x)+x y(2) = 8, x>0
X'
(3)
ii. Hence or otherwise find y(x) at x = 8
(2)
(b) Solve the following initial value problems:
y'(x) + 20 xy(x) + 1 = 0, y(0) = 20, x 2: O
(5)
(c) 1. Suppose a returning student brings the flu virus to his/her boarding house
college campus of 5,000 students. Suppose further that the rate at which the
virus spreads is proportional not only to the number of infected students but
also to the number of students not infected. Determine the number of infected
students after 7 days if it is observed that after 5 days the number of infected
students is 70.
(5)
11. A tank initially holds 300 liters of liquid, with 20 grams of dissolved salt. Brine,
with a salt concentration of 1 gram per liter, is continuously pumped into the
tank at a rate of 4 liters per minute. Simultaneously, a well-mixed solution
is pumped out of the tank at the same rate. Determine the function N(t),
representing the number of grams of salt in the tank at time t.
(5)
2. (a) If y1 and y2 are two solutions of second order homogeneous differential equation of
the form
y"(x) + p(x)y'(x) + q(x)y(x) = J(x)
where p(x) and q(x) are continuous on an open interval I, derive the formula for
u(x) and v(x) by using variation of parameters.
(6)
(b) If
Y1(x) = 2x + 1, W(Y1, Y2) = 2x2 + 2x + 1, Y2(0) = 0
find Y2(x)
(7)
(c) Solve
8x2y"(x) + 16xy'(x) + 2y(x) = 0
(7)
3. Solve the following using Laplace Transform
(a)
2
d y(x)
+ 2dy(t)
+ 5y(t)
=
e-t sin t
dt2
dt
'
y(0) = 0,
y'(0) = 1
(b)
dxd~t)- y(t) = et, dy(t) + X (t) = Sl.n t, X (0) = 1, y (O) = 0
(7)
2
(7)
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ODE 602S
Ordinary Differential Equations
January 2024
(c) If J(t) = t and g(t) = ebt
i. Find the convolution of J(t) and g(t)
(4)
ii. Find the Laplace transform of f(t) ® g(t)
(2)
4. (a) Solve the Euler equation
x2y"(x) + l5xy'(x) + 58y(x) = 0, y(l) = 1, y'(l) = 0
(7)
(b) Solve the following differential equation by method of variation of parameters:
y"(x) + y(x) = tanx
(8)
(c) Solve the following differential equation by method of undetermined coefficients:
y"(x) + 2y'(x) + 2y(x) = -ex(5x - 11), y(0) = -1, y'(0) = -3
(5)
5. (a) Find at least the first four nonzero terms in a power series expansion about x 0 for
the general solution of the following ordinary differential equation at x 0
x2y"(x) - 2xy"(x) + 2y(x) = 0, Xo= 1
(10)
(b) Solve by using Frobenius method:
x2y"(x) + x( x - ~)y'(x) + ~y(x) = 0, at xo = 0
(10)
End of Exam!
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