APM1514/102/3
Study Unit 2 EXERCISES
2.1 For each of the following difference equations, classify the equation as either autonomous or not, and as first-order
or not.
(a) +1 = 2 +
(b) +1 = + (−1 )
(c) +1 =
(d) +1 = 2−1
(e) +1 = + −1 + −2
(f) +1 = 2 + ( )
2.2 Write down 0 1 2 3 and 4 when the difference equation and initial value are as given below.
(a) +1 = 2 0 = 1
(b) +1 = 2 + 8 0 = 0
(c) +1 = 2 ( + 3) 0 = 4
√
(d) +1 = 0 = 2
2.3 Write out terms 0 1 and 2 for the following dynamical systems.
(a) +1 = 2 − 1 0 = 1
(b) +1 = 2 + 2 0 = 0
(c) +1 = 2 − 3 0 = 24
2.4 Find the general solutions to the following difference equations.
(a) +1 = 2
(b) +1 = − 8
√
(c) +1 = 0 = 2
2.5 Find the solution, that is, an expression for as a function of when the difference equation and initial value are
as given below.
(a) +1 = 2 0 = 1
(b) +1 = − 8 0 = 0
√
(c) +1 = 0 = 2
2.6 For the following systems, find the equilibrium point, if one exists.
(a) +1 = 11
(b) +1 = −07
(c) +1 = 2 − 4
(d) +1 =
(e) = 10
2.7 For the following systems, find the equilibrium point(s), if they exist.
(a) +1 = 151
(b) +1 = −09 + 1
(c) +1 = 2 − 4 − 2
(d) +1 = 2
√
(e) +1 =
2.8 Find the equilibrium values of and the general solutions to the following difference equations:
(a) +1 − = 5
3
, (b) +1 = 2
(c) +1 = ( − 1)2
2.9 Find the equilibrium points for all possible values of and for the system
+1 = +
2.10 Find the solution and the equilibrium points for all possible values of for the system
+1 = +
2.11 Describe the outcome of the general linear system
+1 =
if we assume that 0 is a constant. Investigate all possible negative values of . You may assume that 0 0
2.12 Write down the solutions to the following linear difference equations.
(a) +1 = −2
(b) +1 = 20
(c) +1 =
(d) +1 = 01
(e) +1 = − 12
2.13 In each of the difference equations given below, with the given initial value, what is the outcome of the solution as
increases?
(a) +1 = 0 = 1
(b) +1 = 10 0 = 0
3
(c) +1 = 2 0 = −1
(d) +1 = 23 0 = −1
1
(e) +1 = 2 0 = 0
(f) +1 = 099 · 0 = 2
2.14 A difference equation is given by
+1 = 3
(a) If 0 = 2 find 1 and 2
(b) Find all the equilibrium values of the difference equation.
(c) Write down an expression for 1 2 and 3 as a function of the initial value 0
(d) From (c) above, find the general solution to the difference equation.
2.15 Find the equilibrium points of the system
+1 = −
for all possible values of and
2.16 Consider the difference equation
+1 = 1 −
(a) If 0 = 2 find 1 and 2
(b) Find all the equilibrium values of the difference equation.
(c) Find the general solution to the difference equation.
2.17 A difference equation is given by
+1 = 3
(a) If 0 = 2 find 1 and 2
(b) Find all the equilibrium values of the difference equation.
(c) Is = 3
0 the general solution to the difference equation? Justify your answer!
4
, APM1514/102/3
Study Unit 3 EXERCISES
3.1 Write down the difference equations for the following proportional population models, when time is measured in
years.
(a) A population with 2 000 births and 800 deaths 10 000 individuals per year.
(b) A population with 0.002 births and 0.01 deaths per population member per year.
3.2 For each of the following birth and death rates, write down the difference equation of the corresponding, proportional
growth model.
(a) Birth rate = 02 death rate = 005
(b) Birth rate = 001 death rate = 01
(c) Birth rate = 20 death rate = 10
(d) Birth rate = 02 death rate = 02
3.3 For each of the following birth and death rates, write down the solution to the corresponding, proportional growth
model with initial value (0) = 1000 and explain what the outcome will be as increases without bound.
(a) Birth rate = 01 death rate = 001
(b) Birth rate = 02 death rate = 01
(c) Birth rate = 10 death rate = 15
(d) Birth rate = 01 death rate = 01
3.4 Assume that a proportionally growing population has the death rate = 01. What should the birth date be, so that
the difference equation describing the growth of the population is
( + 1) = 095 · ()?
3.5 The size of a proportionally growing population at time is given by
= 100 · (1015)
with measured in years. If the birth rate was 20 births per thousand individuals per year, how many deaths were
there in the population per thousand individuals per year?
3.6 Assume that a population has a birth rate of 3.5 per year and a death rate of 1.5 per year. Find the size of the
population after 1, 2 and 5 years, if the initial population size was
(a) 1000
(b) 10 000
3.7 A population grows according to the difference equation
+1 = 072 ·
with an initial value of 20 000.
(a) How big is the population after 5 years?
(b) How many years does it take for the population to reach 2 000?
3.8 A population grows according to the difference equation
+1 = 145 ·
with an initial value of 10 000.
(a) How long does it take for the population to double in size?
(b) How many years does it take for the population to reach 100 000?
3.9 Write down the difference equation to model the following population: The birth rate is = 08 per person per year,
the death rate is = 02 per person per year, and each year 30 percent of the existing population moves out.
3.10 Write down the difference equation to model the following population: The birth rate is = 01 per person per year,
the death rate is = 03 per person per year, and each year 3000 new individuals move into the population.
3.11 Write down the difference equation to model the following population: The birth rate is = 005 per person per
year, the death rate is = 001 per person per year, and each year 100 population members leave the population.
5
, 3.12 Write down the difference equation to model the following population: The birth rate is = 10 per person per
year, the death rate is = 05 per person per year, and each year every population member persuades one additional
individual to enter the population.
3.13 Consider the following population model with immigration:
+1 = 102 · + 1000
Here, denotes the size of the population after years. Assume that the initial population size, in year = 0 is
10 000.
(a) Calculate the value of 1 from the value of 0 the value of 2 from 1 and the value of 3 from 2
(b) Use the values you calculated in (a) to prove that the solution to the difference equation is NOT given by
= (102) 0 + ∗ 1000 [Finding and using the correct solution does not form a part of this module – if you
are interested, it is given later on in this study unit, in the part dealing with the cheque account model!]
3.14 Assume that the population of a country has a birth rate is = 11 per person per year, and a death rate of = 05
per person per year.
(a) Assume that 1000 people move out of the country each year. What should the initial population be, to ensure that
the population will always increase?
(b) Assume that instead, percent of the population present at the end of each year moves out. What should be to
guarantee that for any initial population size, the population will forever stay constant?
3.15 Write down the difference equations modelling the following investment accounts. Use to denote the amount of
money at the end of month
(a) An account paying interest at the rate of 2% for the amount on the account during each month, with interest paid
at the end of each month.
(b) An account paying interest at the rate of 10% for the amount on the account during each month, with interest
paid at the end of each month.
(c) An account paying interest at the rate of 11% for the amount on the account during each month, with interest
paid at the end of each month.
3.16 Write down the solutions to the difference equations in the previous question, 3.15 — that is, an expression for
in terms of and the initial deposit 0
3.17 An investment account pays interest at the rate of 2% per month. If the initial deposit was R10 000, how much
money will there be in the account
(a) after 3 months,
(b) after 6 months,
(c) after 24 months?
3.18 Assume that I deposit R50 000 into an investment account which pays interest at the rate of 5% per month. How
many months will it take until I will I have more than R60 000 on the account?
3.19 Assume that I deposit R20 000 into an investment account. How much money will I have on the account after 6
months,
(a) if the interest rate is 1% per month,
(b) if the interest rate is 5% per month,
(c) if the interest rate is 10% per month?
3.20 Assume that I deposit R20 000 into an investment account. How many months will it take until I will I have more
than R25 000 on the account,
(a) if the interest rate is 1% per month,
(b) if the interest rate is 5% per month,
(c) if the interest rate is 10% per month?
3.21 An amount of R15 000 is deposited into a bank account, which pays interest at the rate of 5% per month. The amount
of R500 is withdrawn from the account at the end of each month. Calculate the money in the bank account at the end
of month one, month two and month 3.
6
Study Unit 2 EXERCISES
2.1 For each of the following difference equations, classify the equation as either autonomous or not, and as first-order
or not.
(a) +1 = 2 +
(b) +1 = + (−1 )
(c) +1 =
(d) +1 = 2−1
(e) +1 = + −1 + −2
(f) +1 = 2 + ( )
2.2 Write down 0 1 2 3 and 4 when the difference equation and initial value are as given below.
(a) +1 = 2 0 = 1
(b) +1 = 2 + 8 0 = 0
(c) +1 = 2 ( + 3) 0 = 4
√
(d) +1 = 0 = 2
2.3 Write out terms 0 1 and 2 for the following dynamical systems.
(a) +1 = 2 − 1 0 = 1
(b) +1 = 2 + 2 0 = 0
(c) +1 = 2 − 3 0 = 24
2.4 Find the general solutions to the following difference equations.
(a) +1 = 2
(b) +1 = − 8
√
(c) +1 = 0 = 2
2.5 Find the solution, that is, an expression for as a function of when the difference equation and initial value are
as given below.
(a) +1 = 2 0 = 1
(b) +1 = − 8 0 = 0
√
(c) +1 = 0 = 2
2.6 For the following systems, find the equilibrium point, if one exists.
(a) +1 = 11
(b) +1 = −07
(c) +1 = 2 − 4
(d) +1 =
(e) = 10
2.7 For the following systems, find the equilibrium point(s), if they exist.
(a) +1 = 151
(b) +1 = −09 + 1
(c) +1 = 2 − 4 − 2
(d) +1 = 2
√
(e) +1 =
2.8 Find the equilibrium values of and the general solutions to the following difference equations:
(a) +1 − = 5
3
, (b) +1 = 2
(c) +1 = ( − 1)2
2.9 Find the equilibrium points for all possible values of and for the system
+1 = +
2.10 Find the solution and the equilibrium points for all possible values of for the system
+1 = +
2.11 Describe the outcome of the general linear system
+1 =
if we assume that 0 is a constant. Investigate all possible negative values of . You may assume that 0 0
2.12 Write down the solutions to the following linear difference equations.
(a) +1 = −2
(b) +1 = 20
(c) +1 =
(d) +1 = 01
(e) +1 = − 12
2.13 In each of the difference equations given below, with the given initial value, what is the outcome of the solution as
increases?
(a) +1 = 0 = 1
(b) +1 = 10 0 = 0
3
(c) +1 = 2 0 = −1
(d) +1 = 23 0 = −1
1
(e) +1 = 2 0 = 0
(f) +1 = 099 · 0 = 2
2.14 A difference equation is given by
+1 = 3
(a) If 0 = 2 find 1 and 2
(b) Find all the equilibrium values of the difference equation.
(c) Write down an expression for 1 2 and 3 as a function of the initial value 0
(d) From (c) above, find the general solution to the difference equation.
2.15 Find the equilibrium points of the system
+1 = −
for all possible values of and
2.16 Consider the difference equation
+1 = 1 −
(a) If 0 = 2 find 1 and 2
(b) Find all the equilibrium values of the difference equation.
(c) Find the general solution to the difference equation.
2.17 A difference equation is given by
+1 = 3
(a) If 0 = 2 find 1 and 2
(b) Find all the equilibrium values of the difference equation.
(c) Is = 3
0 the general solution to the difference equation? Justify your answer!
4
, APM1514/102/3
Study Unit 3 EXERCISES
3.1 Write down the difference equations for the following proportional population models, when time is measured in
years.
(a) A population with 2 000 births and 800 deaths 10 000 individuals per year.
(b) A population with 0.002 births and 0.01 deaths per population member per year.
3.2 For each of the following birth and death rates, write down the difference equation of the corresponding, proportional
growth model.
(a) Birth rate = 02 death rate = 005
(b) Birth rate = 001 death rate = 01
(c) Birth rate = 20 death rate = 10
(d) Birth rate = 02 death rate = 02
3.3 For each of the following birth and death rates, write down the solution to the corresponding, proportional growth
model with initial value (0) = 1000 and explain what the outcome will be as increases without bound.
(a) Birth rate = 01 death rate = 001
(b) Birth rate = 02 death rate = 01
(c) Birth rate = 10 death rate = 15
(d) Birth rate = 01 death rate = 01
3.4 Assume that a proportionally growing population has the death rate = 01. What should the birth date be, so that
the difference equation describing the growth of the population is
( + 1) = 095 · ()?
3.5 The size of a proportionally growing population at time is given by
= 100 · (1015)
with measured in years. If the birth rate was 20 births per thousand individuals per year, how many deaths were
there in the population per thousand individuals per year?
3.6 Assume that a population has a birth rate of 3.5 per year and a death rate of 1.5 per year. Find the size of the
population after 1, 2 and 5 years, if the initial population size was
(a) 1000
(b) 10 000
3.7 A population grows according to the difference equation
+1 = 072 ·
with an initial value of 20 000.
(a) How big is the population after 5 years?
(b) How many years does it take for the population to reach 2 000?
3.8 A population grows according to the difference equation
+1 = 145 ·
with an initial value of 10 000.
(a) How long does it take for the population to double in size?
(b) How many years does it take for the population to reach 100 000?
3.9 Write down the difference equation to model the following population: The birth rate is = 08 per person per year,
the death rate is = 02 per person per year, and each year 30 percent of the existing population moves out.
3.10 Write down the difference equation to model the following population: The birth rate is = 01 per person per year,
the death rate is = 03 per person per year, and each year 3000 new individuals move into the population.
3.11 Write down the difference equation to model the following population: The birth rate is = 005 per person per
year, the death rate is = 001 per person per year, and each year 100 population members leave the population.
5
, 3.12 Write down the difference equation to model the following population: The birth rate is = 10 per person per
year, the death rate is = 05 per person per year, and each year every population member persuades one additional
individual to enter the population.
3.13 Consider the following population model with immigration:
+1 = 102 · + 1000
Here, denotes the size of the population after years. Assume that the initial population size, in year = 0 is
10 000.
(a) Calculate the value of 1 from the value of 0 the value of 2 from 1 and the value of 3 from 2
(b) Use the values you calculated in (a) to prove that the solution to the difference equation is NOT given by
= (102) 0 + ∗ 1000 [Finding and using the correct solution does not form a part of this module – if you
are interested, it is given later on in this study unit, in the part dealing with the cheque account model!]
3.14 Assume that the population of a country has a birth rate is = 11 per person per year, and a death rate of = 05
per person per year.
(a) Assume that 1000 people move out of the country each year. What should the initial population be, to ensure that
the population will always increase?
(b) Assume that instead, percent of the population present at the end of each year moves out. What should be to
guarantee that for any initial population size, the population will forever stay constant?
3.15 Write down the difference equations modelling the following investment accounts. Use to denote the amount of
money at the end of month
(a) An account paying interest at the rate of 2% for the amount on the account during each month, with interest paid
at the end of each month.
(b) An account paying interest at the rate of 10% for the amount on the account during each month, with interest
paid at the end of each month.
(c) An account paying interest at the rate of 11% for the amount on the account during each month, with interest
paid at the end of each month.
3.16 Write down the solutions to the difference equations in the previous question, 3.15 — that is, an expression for
in terms of and the initial deposit 0
3.17 An investment account pays interest at the rate of 2% per month. If the initial deposit was R10 000, how much
money will there be in the account
(a) after 3 months,
(b) after 6 months,
(c) after 24 months?
3.18 Assume that I deposit R50 000 into an investment account which pays interest at the rate of 5% per month. How
many months will it take until I will I have more than R60 000 on the account?
3.19 Assume that I deposit R20 000 into an investment account. How much money will I have on the account after 6
months,
(a) if the interest rate is 1% per month,
(b) if the interest rate is 5% per month,
(c) if the interest rate is 10% per month?
3.20 Assume that I deposit R20 000 into an investment account. How many months will it take until I will I have more
than R25 000 on the account,
(a) if the interest rate is 1% per month,
(b) if the interest rate is 5% per month,
(c) if the interest rate is 10% per month?
3.21 An amount of R15 000 is deposited into a bank account, which pays interest at the rate of 5% per month. The amount
of R500 is withdrawn from the account at the end of each month. Calculate the money in the bank account at the end
of month one, month two and month 3.
6
