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1. An equation used to model (self-limiting) population growth (such as the growth of tumours)
is
dP K
= rP ln ,
dt P
where P (t) > 0 is the population at time t ≥ 0, and r and K are positive constants.
(a) Find all equilibrium solutions of this differential equation and classify each as stable,
unstable, or semi-stable.
(5 marks)
(b) Generate a direction field for the case r = 0.5 and K = 500 for 0 ≤ t ≤ 50 and
1 ≤ P ≤ 700. Clearly state how your direction field confirms your statements in part
(a) about the value and stability of each equilibrium solution.
(5 marks)
(c) Solve this differential equation along with intial condition P (0) = P0 , giving P as an
explicit function of t in your final answer.
(15 marks)
[25 MARKS TOTAL]
4
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2. (a) Solve the following differential equation:
dy 4x − 2xy 3 sin(x2 y 3 )
=
dx 3x2 y 2 sin(x2 y 3 )
Hint: you do not have to write y as an explicit function of x in your final answer.
(12 marks)
(b) For the initial value problem
dy
= −y 2 (y − 2)2 (y + 3), y(0) = y0 (where y0 ∈ R),
dt
determine the value of all equilibrium points and state, with reason, whether each equi-
librium point is stable, unstable, or semistable.
(13 marks)
[25 MARKS TOTAL]
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3. A small tank with a capacity of 1000 litres originally contains 200 litres of water with 100
kilograms of salt in solution. Water containing 1 kilogram of salt per litre starts entering the
tank at a rate of 3 litres per minute and the mixture is at the same time allowed to flow out of
the tank at a rate of 2 litres per minute.
(a) Give a formula for the amount of solution in the tank at time t minutes after the inflow
and outflow begin, and calculate at what time the solution would begin to overflow the
tank.
(4 marks)
(b) If Q(t) is the amount of salt in the tank (in kilograms) at time t minutes after the inflow
and outflow begin, set up an initial value problem whose solution would allow us to know
Q(t) at any time, t, prior to the time when the solution begins to overflow the tank. Solve
this initial value problem for Q(t).
(14 marks)
(c) What is the amount of salt (in kilograms) AND the concentration (in kilograms per litre)
of salt in the tank when the solution first begins to overflow the tank? (4 marks)
(d) What is the theoretical limiting concentration of salt in the tank if the tank had infinite
capacity? Explain your answer.
(3 marks)
[25 MARKS TOTAL]
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4. (a) Solve the Initial Value Problem
d2 x
+ 6x = 12, x(0) = 1, x0 (0) = 14
dt2
(12 marks)
(b) One of the (many) types of differential equations we have not considered in class is the
unfortunately named (due to potential confusion with the homogeneous linear differential
equations which are 0 on one side of the equation) homogeneous first order differential
equation which, is a differential equation which can be written in the form
dy y
=f .
dx x
Research this type of homogeneous differential equation and use the knowledge you gain
to solve the following homogeneous differential equation (your final answer should be an
equation giving y implicitly as a function of x):
dy 3y 2 − x2
=
dx 2xy
(13 marks)
[25 MARKS TOTAL]
7
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1. An equation used to model (self-limiting) population growth (such as the growth of tumours)
is
dP K
= rP ln ,
dt P
where P (t) > 0 is the population at time t ≥ 0, and r and K are positive constants.
(a) Find all equilibrium solutions of this differential equation and classify each as stable,
unstable, or semi-stable.
(5 marks)
(b) Generate a direction field for the case r = 0.5 and K = 500 for 0 ≤ t ≤ 50 and
1 ≤ P ≤ 700. Clearly state how your direction field confirms your statements in part
(a) about the value and stability of each equilibrium solution.
(5 marks)
(c) Solve this differential equation along with intial condition P (0) = P0 , giving P as an
explicit function of t in your final answer.
(15 marks)
[25 MARKS TOTAL]
4
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2. (a) Solve the following differential equation:
dy 4x − 2xy 3 sin(x2 y 3 )
=
dx 3x2 y 2 sin(x2 y 3 )
Hint: you do not have to write y as an explicit function of x in your final answer.
(12 marks)
(b) For the initial value problem
dy
= −y 2 (y − 2)2 (y + 3), y(0) = y0 (where y0 ∈ R),
dt
determine the value of all equilibrium points and state, with reason, whether each equi-
librium point is stable, unstable, or semistable.
(13 marks)
[25 MARKS TOTAL]
5
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3. A small tank with a capacity of 1000 litres originally contains 200 litres of water with 100
kilograms of salt in solution. Water containing 1 kilogram of salt per litre starts entering the
tank at a rate of 3 litres per minute and the mixture is at the same time allowed to flow out of
the tank at a rate of 2 litres per minute.
(a) Give a formula for the amount of solution in the tank at time t minutes after the inflow
and outflow begin, and calculate at what time the solution would begin to overflow the
tank.
(4 marks)
(b) If Q(t) is the amount of salt in the tank (in kilograms) at time t minutes after the inflow
and outflow begin, set up an initial value problem whose solution would allow us to know
Q(t) at any time, t, prior to the time when the solution begins to overflow the tank. Solve
this initial value problem for Q(t).
(14 marks)
(c) What is the amount of salt (in kilograms) AND the concentration (in kilograms per litre)
of salt in the tank when the solution first begins to overflow the tank? (4 marks)
(d) What is the theoretical limiting concentration of salt in the tank if the tank had infinite
capacity? Explain your answer.
(3 marks)
[25 MARKS TOTAL]
6
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4. (a) Solve the Initial Value Problem
d2 x
+ 6x = 12, x(0) = 1, x0 (0) = 14
dt2
(12 marks)
(b) One of the (many) types of differential equations we have not considered in class is the
unfortunately named (due to potential confusion with the homogeneous linear differential
equations which are 0 on one side of the equation) homogeneous first order differential
equation which, is a differential equation which can be written in the form
dy y
=f .
dx x
Research this type of homogeneous differential equation and use the knowledge you gain
to solve the following homogeneous differential equation (your final answer should be an
equation giving y implicitly as a function of x):
dy 3y 2 − x2
=
dx 2xy
(13 marks)
[25 MARKS TOTAL]
7
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