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Department of Mathematics

MATHS

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08 Assignment 3 Due: 4pm, Tuesday 4 June 20

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Hand your assignment in to the correct box in the Basement of Building 303, before the due date.
Please use a Department of Mathematics Cover Sheet, available from the Basement of Building
303. Show all working and note that late assignments will not be marked.

This assignment was written by Sione Ma‘u (s.mau@auckland.ac.nz). The tutors in the Mathemat-
ics Assistance Room (on the Ground Floor of Building 303) can give you help and advice on the
concepts covered, but all work submitted must be your own.

1. (20 marks)

The black robbin is the most rare bird in the world. It is found only in the Chatham Islands,
New Zealand. The population of black robbins changes on a yearly basis as a discrete dynamic
system. Suppose that initially there are 30 juvenile chicks and 20 breeding adults, that is

x0 =

(
30
20

)
. Suppose also that the yearly transition matrix is A =

(
0 1.25

0.75 0.5

)
.

(a) Which entry in the transition matrix gives the annual birthrate of chicks per adult?

(b) Which entry tells us the proportion of chicks that will survive to become adults?

(c) Calculate the state vector x1.

(d) Find the eigenvalues and corresponding eigenvectors for matrix A.

(e) Express x0 as a linear combination of eigenvectors.

(f) Find the long-term distribution, after n years. State the long term ratio of juveniles to
adults.

(g) From your result to part (f), is the black robbin likely to survive?

2. (20 marks, 4 marks each)

(a) State the order of the following DE, and confirm that the functions in the given family
are solutions

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dy

dx
+ y = x − 1; y = ce−

x
2 + x − 3 (c a constant).

(b) Use implicit differentiation to confirm that the equation x2 +xy2 = C defines an implicit
solution of the differential equation

2x + y2 + 2xy
dy

dx
= 0.

CONTINUED

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(c) Solve the separable linear first order DE

dy

dx
− 4xy = 0,

using both the method of separation of variables and using an integrating factor.

(d) Use an integrating factor to solve

(x2 + 1)
dy

dx
+ xy = 0.

(e) Solve the initial value problem

y′ − xey = 2ey, y(0) = 0.

3. (10 marks)

Doctors are studying the spread of measles. Using the variable y to represent the proportion
of a population affected by measles, and using t to represent time (days) they find the rate
of spread of measles through a city can be modeled by the differential equation

dy

dt
= 0.2(0.6 − y).

Suppose initially ten percent of a cities population has measles.

(a) Solve this IVP (initial value problem).

(b) Sketch a graph showing your solution to this IVP.

(c) How long will it take before half the population are affected by measles?

(d) What is the long term proportion of the population which will be affected by measles?

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