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ODE602S - ORDINARY DIFFERENTIAL EQUATIONS - 2ND OPP - JAN 2023 |
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nAm I BI A un IVE RSITY
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: JANUARY 2023
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 80
SUPPLEMENTARY/ SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
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
January 2023
1. Solve the following initial value problems:
(a)
x2y'(x) 3
+xy(x)=
5
e-x
5
,
y(-1)=0,
for x<0
(5)
(b)
y'(x) sinx + y(x) cosx = 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
d:?) = P(t)(5 - P(t)) - 4, P(0) = Po
i. Solve the IVP.
(5)
11. Determine the time when the fishery population becomes quarter 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) Solve the Euler equation
x 2y"(x) + 15xy'(x) + 58y(x) = 0, y(l) = 1, y'(l) = 0
(7)
(b) Solve the following differential equations by method of variation of parameters
y"(x) + y(x) = tanx
(8)
(c) Solve the following differential equations by method of undetermined coefficients
y"(x) + 2y'(x) + 2y(x) = -ex(5x - 11), y(0) = -1, y'(0) = -3
(5)
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ODE 602S
Ordinary Differential Equations
4. (a) Find the Laplace inverse of
s2 - 10s + 13
(s-7)(s 2 -5s+6)
(b) Compute
[, 1 { 2s3 + 2s2 + 4s + 1 }
(s2 + 1)(s2 + s + 1)
(c) Solve using Laplace transform
y'(t) - 2y(t) = 6t3e2t,
y(0) = -3
January 2023
(6)
(7)
(7)
5. (a) Use reduction of order method to find Y2(x) if
x 2y" - 3xy' + 4y = O; Y1(x) = x2
(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) Use the power series method to solve
y"(x) + 4y(x) = 0, y(0) = 1, y'(0) = 2
(10)
End of Exam!