The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH2021: Vector Calculus and Differential Equations Semester 1, 2024
Lecturers: Pieter Roffelsen and Haotian Wu
This individual assignment is due by 11:59pm Thursday 16 May 2024, via Canvas. Late assignments
will receive a penalty of 5% of the maximum mark for each calendar day after the due date. After ten
calendar days late, a mark of zero will be awarded.
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include your name in any area of your assignment; only your SID should be present.
The School of Mathematics and Statistics encourages some collaboration between students when working
on problems, but students must write up and submit their own version of the solutions.
This assignment is worth 10% of your final assessment for this course. Your answers should be well written,
neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any resources used and show all
working. Present your arguments clearly using words or explanations and diagrams where relevant. After all,
mathematics is about communicating your ideas. This is a worthwhile skill which takes time and effort to
master.
Copyright © 2024 The University of Sydney 1
1. (Change of variables) Consider the homogeneous 2nd order ODE
′′ − 1
′ + 16 2 = 0, (1)
on the open interval I = (0,∞). You are going to use the change of variables ↦→ , with
= 2, to study this ODE.
(a) Write () = (2) and compute the first and second derivative of in terms of .
Substitute the results into differential equation (1) and show that satisfies
′′() + 4 () = 0, (2)
where = 2.
(b) Construct a fundamental set of solutions {1, 2} of differential equation (2).
(c) Use the result in part (b) to derive a fundamental set of solutions {1, 2} of differential
equation (1) and compute its Wronskian.
(d) Find a particular solution to the inhomogeneous ODE
′′ − 1
′ + 16 2 = 4 4.
2. (Reconstruction of an ODE) A group of scientists at USYD have done countless experiments,
which all point to a single homogeneous linear 2nd order ODE on I = R,
′′ + ()′ + () = 0.
However, they have not been able to determine the coefficients () and () of the differential
equation. Their experiments indicate that there exist solutions 1() and 2() such that
• the Wronskian of 1 and 2 is identically equal to 1,
• 1 is also a solution to the 1st order differential equation ′ + sin() = 0.
Can you help the scientists find the coefficients () and ()?
Hint: first use Abel’s formula to find ().
3. (Airy’s equation) You have travelled to the early nineteenth century and astronomer George
Airy has asked for your help. George Airy has developed a model for supernumerary rainbows1,
involving the following 2nd order ordinary differential equation,
′′ = .
Your task is to find the unique solution to this differential equation satisfying the initial condi-
tions (0) = 0 and ′(0) = 1.
(a) Substitute a power series () =
∞∑︁
=0
around = 0 into the differential equation
and derive a recurrence relation for the coefficients.
(b) Use the initial conditions and recurrence relation to find the first three non-zero terms
in the power series.
1For optional background reading, see The Mathematics of the Rainbow, Part II