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Chapter 12
Systems of Dierential Equations
Exercises
1. Two species with populations x and y compete for their food supply. The equations describing the evolution of x and y are
x = ax by
y = cx + dy ,
where a, b, c, d are positive constants. Explain w
14
Chapter 8
Models and Dierential Equations
Exercises
1. Find the general solutions and sketch the solution curves of the dierential equations:
(a)
dy
= ex ,
dx
(b)
dy
= sin x ,
dx
(c)
dy
= sinh x .
dx
Answers. General solutions are:
(a) y = ex + C ,
(b)
9
Chapter 6
Integration Techniques: I
Exercises
1. Use substitutions to evaluate the following integrals
(a)
(c)
x1 log x dx
e2x
dx
ex + 1
x
(b)
e
dx
(d)
1 + x2
dx .
x4
Solution.
(a) Let u = log x. Then du = u (x) dx = x1 dx. Substituting,
x1 log x dx =
(
7
Chapter 5
Integrals as Functions
Exercises
1. When applying the Fundamental Theorem it is important to check that the conditions for the theorem are satised. In particular, discontinuities in the function
or its derivative can invalidate the formula. Co
11
Chapter 7
Integration Techniques: 2
Exercises
1. Find a reduction formula for the indenite integral
In =
dx
.
(1 + x2 )n
Hint: Take dv/dx = 1.
Solution. The case n = 1 is a standard integral: I1 =
dx
= tan1 x + C.
(1 + x2 )
For any n integrate by parts
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Chapter 10
Linear Dierential Equations
Exercises
1. Find the general solutions of
dx
tx = t ,
dt
dy
(d)
+ 2y = ex ,
dx
dy
(f) x2
+ (1 2x)y = x2 .
dx
dy
2y = 3 ,
dx
dy
4x3 y
(c)
=
,
dx
x
dx
(e)
+ 2tx = 2t3 ,
dt
(b)
(a)
Answers. General solutions are
2
20
dx
= 0.10x + 1000,
dt
(b) 23 years.
(a)
12. The spread of innovation (in agriculture and industry) has been successfully modelled by assuming that the rate of spread is proportional to both the number of
people already having adopted the new system and
17
Chapter 9
Applications of Separable Equations
Exercises
1. Fechners law states that
dR
n
= .
dS
S
Determine R(S).
(An example of its application to describe the response of the eye to brightness
is given in Table 1.)
Answer. R = n ln S + C . The visual
24
(b) c(t) = c0 ekt .
10. In a dilute sugar solution the rate of decrease of concentration is proportional to
the concentration c. If c = 0.01 g/cm3 at time t = 0 and c = 0.005 g/cm3 at time
t = 4 hours, show that the concentration after 10 hours will 0.
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Chapter 11
Second-Order Dierential Equations
Exercises
1. Find the general solution of
d4 y
= 0.
dx4
What is the particular solution satisfying y(0) = y (0) = y (0) = y (0) = 1?
Answer.General solution: y = Ax3 + Bx2 + Cx + D.
1
1
Particular solution:
1
Chapter 1
Introduction
Exercises
1. A mothball initially has radius 0.5 cm and slowly evaporates.
(a) If V denotes the volume of the mothball and r the radius, use the chain rule
for dierentiation to show that
dV
dr
= 4r2 .
dt
dt
(b) Suppose that the ra