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ODE602S - ORDINARY DIFFERENTIAL EQUATIONS - 1ST OPP - NOV 2022 |
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n Am I BI A u n IVER s ITY
OF SCIEnCE Ano TECHnOLOGY
FACULTY OF HEALTH, NATURAL RESOURCES AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of science; Bachelor of science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSOC; 07BSAM
LEVEL: 6
COURSE CODE: ODE602S
COURSE NAME: ORDINARYDIFFERENTIAL
EQUATIONS
SESSION: NOVEMBER 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 80
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Prof A.S EEGUNJOBI
MODERATOR:
Prof S.A REJU
INSTRUCTIONS
1. Answer ANY FOUR(4) questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
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ODE 602S
Ordinary Differential Equations
November 2022
l. Solve the following initial value problems:
(a) x 2y'(x) + 5x3y(x) = e-x, y(-1) = 0, for x < 0
(5)
(b) sinxy'(x) + cosxy(x) = 2ex, y(l) = a, 0 < x < 1r
(5)
(c) If a constant number k of fish are harvested from a fishery per unit time, then a
logistic model for the population P(t) of the fishery at time t is given by
ddP,t(t) = -P(t)(P(t) - 5) - 4, P(0) = Po
1. Solve the IVP.
(5)
11. Determine the time when the fishery population becomes half of the initial
population
(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) = f(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
find y2 (x)
(7)
(c) Solve
8x2y"(x) + 16xy'(x) + 2y(x) = 0
(7)
3. (a) Find the general solution of
(6)
(b) Find the general solution of
y"'(x) - 6y"(x) + lly'(x) - 6y = e- 2x + e- 3x
(7)
(c) Solve the following differential equations simultaneously
dx
dt
+ 5x(t)
-
2y(t)
=
t,
dy
dt
+ 2x(t)
+ y(t)
=
0
(7)
4. (a) Calculate
.C{9t4 + 6t~}
(6)
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ODE 602S
Ordinary Differential Equations
(b) Using Convolution theorem, find
November 2022
(7)
(c) Solve the following IVP:
2y"(t) - 6y'(t) + 4y(t) = 4e3t, y(0) = 5, y'(0) = 7
(7)
5. (a) Find the radius of convergence of the following power series
(3n)! n
6 (n!)3x
n=O
(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) Find series solution of IVP
5y"(x) + lOxy'(x) + 5(1 + x2 )y(x) = 0, with y(0) = 3, y'(0) = -1.
when the expansion is about the origin.
End of Exam!