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MATH2021: Vector Calculus and Differential Equations
创建日期：2024-05-08 14:50:02
所属分类：数学mathematics
标签： MATH2021
Vector Calculus and Differential Equations

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MATH2021: Vector Calculus and Differential Equations 

This individual assignment is due by 11:59pm Thursday 12 May 2022, via
Canvas. Late assignments will receive a penalty of 5% per day until the closing date.
A single PDF copy of your answers must be uploaded in Canvas. Please make sure you
review your submission carefully. What you see is exactly how the marker will see your
assignment. To ensure compliance with our anonymous marking obligations, please do
not under any circumstances 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 whenworking on problems, but students must
write up and submit their own version of the solutions.
This assignment is worth 5% 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 of explanation
and diagrams where relevant. After all, mathematics is about communicating your ideas. This
is a worthwhile skill which takes time and effort to master. The marker will give you feedback
and allocate an overall letter grade and mark to your assignment using the following criteria:
Mark Grade Criterion
8 A Outstanding and scholarly work, answering all parts correctly, with clear
accurate explanations and all relevant diagrams and working. There are
at most only minor or trivial errors or omissions.
7 B Very good work, making excellent progress, but with one or two substantial
errors, misunderstandings or omissions throughout the assignment.
6 C Good work, making good progress, but making more than two distinct
substantial errors, misunderstandings or omissions throughout the assign-
ment.
5 D A reasonable attempt, but making more than three distinct substantial
errors, misunderstandings or omissions throughout the assignment.
3 E Some attempt, with limited progress made.
1 F Extremely limited attempt.
0 Z No credit awarded.
Copyright c© 2022 The University of Sydney 1
The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH2021: Vector Calculus and Differential Equations Semester 1, 2022
Lecturer: James Parkinson, Fernando Viera
Differential Equations
Exact Equations
1. Construct an Exact equation that is satisfied by the function
F (x, y) = cos(x)esin(xy)
2. Show that F (x, y) = x2y + exy (1) is a solution of the equation
y
(
2x + exy
)
dx + x
(
x + exy
)
dy = 0
Separable Equations
3. Which of the following differential equations are Separable? Solve the ones that are
separable:
(a) (1 + x)
dy
dx
+ y2 = 0
(b) y
dy
dx
= (x− y2) sin y
(c)
dy
dx
=
x + 1
2xy
Linear First Order Equations
4. Solve the following Linear First Order Equation using the Integrating Factor Method:
(1 + x2)
dy
dx
= 1− 2xy.
5. Use the method of Variation of Parameters to find the general solution of the following
linear first order equation:
dy
dx
− 3y = xe−x.
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