MATH1023:
题意： 给定x,t的积分关系，根据题目里不同的问题解出两个x(t)，然后在不同条件下比较这两个函数 解析： 首先给出了dx与dt的关系，可以通过这个算式得出积分后的x与t的关系，随后通过给定的条件可以解出x(t)；求出两种题设下的x(t)后，再比较二者
涉及知识点： 不定积分，根据条件解待定系数
The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH1023: Multivariable Calculus and Modelling Semester 2, 2019
Web Page: http://sydney.edu.au/science/maths/u/UG/JM/MATH1023/
Lecturers: Eduardo Altmann and Leo Tzou
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Copyright c 2019 The University of Sydney 1
Let x(t) 2 [0; 1] be the fraction of maximum capacity of a live-music venue at time t (in hours)
after the door opens. The rate at which people go into the venue is modeled by
dx
dt = h(x)(1 - x); (1)
where h(x) is a function of x only.
1. Consider the case in which people with a ticket but outside the venue go into it at a
constant rate h = 1=2 and thus
dx
dt =
1 2
(1 - x):
(a) Find the general solution x(t).
(b) The initial crowd waiting at the door for the venue to open is k 2 [0; 1] of the
maximum capacity (i.e. x(0) = k). How full is the venue at t?
2. Suppose people also decides whether to go into the venue depending on if the place looks
popular. This corresponds to h(x) = 3 2x and thus
dx
dt =
3 2
x(1 - x):
(a) Find the general solution x(t).
(b) What should be the initial crowd x(0) if the band wants to start playing at t = 2
hours with 80% capacity?
3. Consider the two models, A and B, both starting at 10% full capacity. Model A is
governed by the process of question (1) and model B is governed by the process described
in question (2).
Start this question by writing down the respective particular solutions xA(t) and xB(t).
(a) Which of the two models will first reach 50% of full capacity?
(b) Which of the two models will first reach 99% of full capacity?
(c) Plot the curves xA(t) and xB(t). Both curves should be consistent with:
(i) your answers to the two previous items;
(ii) the rate of change at t = 0 (i.e., dx
dt at t = 0);
(iii) the values of x in the limit t ! 1.
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