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Ordinary Differential Equations I Question Paper
Ordinary Differential Equations I
Course:Bachelor Of Science
Institution: Kenyatta University question papers
Exam Year:2009
KENYATTA UNIVERSITY
UNIVERSITY EXAMINATIONS 2009/2010
FIRST SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR OF
ARTS, BACHELOR OF EDUCATION AND BACHELOR OF SCIENCE
SMA 335: ORDINARY DIFFERENTIAL EQUATIONS I
DATE: Monday 21st December, 2009 TIME: 11.00 a.m. – 1.00 p.m.
INSTRUCTIONS
Answer question ONE and any other TWO questions.
Question One – (30 marks)
a)
Given the differential equation
y '+ 2x
y ' 3
+ xy = 0
Determine the order, degree and
linearity.
[2
marks]
b)
Find the third order differential equation associated to the primitive
y = Asin (4x + B) + C e3x
[4 marks]
c)
Show that the differential equation below is exact and hence solve it
(ylny + 2xlnx
)dx + ( xlny x
+ )dy = 0
[4 marks]
d)
Find the solution of the initial value problem
d2 y + ( tan x)dy
=
x
cos
2
dx
dx
[4 marks]
= '
-
y(0) y
(0) =
1,
p x
<
p
<
2
2
e)
Solve the differential equation
D(D – 1)2 (D2 + 2D + 4)y = 0
[3 marks]
f)
Use the method of undetermined coefficients to solve the non-homogeneous
linear differential equation
4x
y '
y
3
-
' 18y
-
xe
=
[5 marks]
Page 1 of 3
g)
A contagious disease is known to spread at a rate of proportional to the number of
people already infected. The infected people initially are 500 and after four days
the number has increased by 30%. Find the time it takes for the number of
infected people to increase to 850.
[4 marks]
h)
Find the solution of the system
dx x
=
+ y
dt
[4 marks]
dy = 4x
+ y
dt
Question Two – (20 marks)
a)
Solve the initial value problem
dy
3x 2 e
-
x
=
,
y(0) =1
dx
2y - 5
Determine the interval on which the solution is valid.
[7 marks]
b)
Show that the differential equation
2
2
x
+ y
y = F(x,
y)
=
is homogeneous
2
2x
Express it in the form y '= (y
g
[6 marks]
x )
c)
A bottle of soda at room temperature (72oF) is placed in a refrigerator where the
temperature is 44oF. After half an hour the soda has cooled to 61oF.
i)
What is the temperature of the soda after another half hour?
[3 marks]
ii)
How long does it take for the soda to cool to 50oF?
[2 marks]
lim
iii)
Find T(t)
(limiting temperature) and sketch the graph of the
t ? 8
temperature
function.
[2
marks]
Question Three – (20 marks)
a)
Solve the differential equation
dy
2x + 3y
+ 2
=
[5 marks]
dx
4x +
3
-
6y
Page 2 of 3
b)
Show that the equation
{1 + (x + y) tan y } dy + dx = 0 has an integrating factor of the form (x + y)n,
where n is a constant. Solve the equation.
[8 marks]
c)
Transform the Bernoulli equation dy
n
+ py
= Qy
dx
into first order, first degree
and hence solve the equation dy
2
2
x
+ y
= y x
dx
, with conditions y(1) = ¼.
[7 marks]
Question four - (20 marks)
a)
Solve the following differential equation
(D4 – 3D3 – 12D2 + 52D – 48)y = 0
[6 marks]
b)
One of the roots to the auxiliary/characteristic roots to the differential equation
(D4 + 2D3 – 7D2 + 2D + 70)y = 0 is 2 -
i
3 . Solve the d.e.
[6 marks]
c)
Given that x, x2, x-1 are the solutions to the auxiliary/homogeneous equation
corresponding to
y '
' + P
' +
' +
=
x >
2 (
x) y P
1(x)y P
(x)y
2x
4 ,
0
0
Determine a particular solution and hence write the general solution to the
d.e. (use the variation of parameters
method). [8
marks]
Question five – (20 marks)
a) i) Explain
the
meaning
of simple harmonic motion.
ii) If
x& + &
x
4
+ 29x
=
cos5t,
find x in terms of t and deduce that when t is large
the motion of the particle is approximately simple harmonic motion and of
period 2p 5 .
[10 marks]
b)
Use power series method to find the general solution of y '
+ y = 0
about the
point x = 0. Show that the solution obtained is a series for cos x and sin x.
[10 marks]
………………….
Page 3 of 3
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