MA3264 TUTORIAL 2
1.
Solve the following differential equations:
(a) xy 0 + (1 + x)y = e−x ,
(b) y 0 − (1 + x3 )y = x + 2,
(c) y 0 + y +
x
y
=0
x>0
y(1) = e − 1,
x>0
(d) 2xyy 0 + (x − 1)y 2 = x2 ex ,
x>0
2. If a cable is held up at two ends at the same height, then it will sag in the middle, making a
U-shaped curve called a catenary. This is the shape seen in electricity cables suspended between
poles, in countries less advanced than Singapore, such as Japan and Australia. It can be shown
using simple physics that if the shape is given by a function y(x), then this function satisfies
dy
µ
=
dx
T
Z
x
r
0
dy 2
+ 1 dt,
dt
where x = 0 at the lowest point of the catenary and y(0) = 0, where µ is the weight per unit length
of the cable, and where T is the horizontal component of its tension; this horizontal component is
a constant along the cable. Find a formula for the shape of the cable. [Hint: use the Fundamental
Theorem of Calculus, and think of the resulting equation as a first-order ODE.]
3. A certain MA3264 student starts a rumour to the effect that one of the professors has been
seen dating Lady Gaga. The number of students who have heard the rumour, R(t), is given by
dR
= KR[45 − R],
dt
where K is a positive constant, and 45 is the number of students taking MA3264. What is the
meaning of K? Is this equation reasonable? [Hint: surely the rumour will spread slowly both when
hardly anyone has heard it yet, but also when nearly everyone has already heard it!] By regarding
this equation as a Bernoulli equation, find R(t).
4. In the lecture notes [Chapter 1] we studied two very simple models of possible constraints on
carbon dioxide emissions: in one case we considered reducing the *amount* of gas being emitted
[example 10A], and in the other case we considered reducing the *concentration* [example 10B].
Ideally one should do both. With that in mind....
A tank contains, at time t = 1 second, 100 cubic metres of pure water. At that instant,
salty water with a concentration of 10/t kilograms per cubic metre begins to be pumped in at a
rate of 10/t cubic metres per second, and the mixed solution is pumped out at the same rate. Find
an expression for the amount of salt in the tank. By sketching a graph of the amount of salt in the
tank as a function of time [feel free to use either Graphmatica or Wolfram Alpha], estimate the
maximum amount of salt in the tank. Compare this with the maxima attained in the other two
cases.
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5. The half-life of Thorium 230 is about 75000 years, while that of Uranium 234 is about 245000
years. A certain sample of ancient coral has a Thorium/Uranium ratio of 10 percent. How old is
the coral?
6.
Recall that the equation governing the spread of the potato blight has the form
dx
= Kf (t)(1 − x),
dt
where f (t) ≡ x(t − p) − x(t − p − q). As in the lectures, we assume that no plant is sick until t = 0,
when suddenly a fraction α get sick. Show that
Z t
x(t) = 1 − (1 − α) exp −K
f (τ )dτ .
0
Well, this doesn’t tell us x(t), because x(t) is on the right side, but we can still use it as follows.
Rt
R t−a
First, show that 0 x(τ − a)dτ = 0 x(τ )dτ for any positive number a. (Remember: x(t) = 0
Rt
R (t−p)
for all negative t.) Use that to prove that 0 f (τ )dτ = (t−p)−q x(τ )dτ. Now use this to prove that
Rt
limt→∞ 0 f (τ )dτ = qβ, where β ≡ limt→∞ x(t), the fraction of plants that get sick in the long
run. (Hint: remember that sick plants never recover, so x(t) is a monotonically increasing function,
which is bounded above. Draw a graph of such a thing, indicating where (t − p) and (t − p) − q are
located on the t axis. Then imagine sliding everything to the right. Pure maths students might
prefer to invoke the mean value theorem for integrals....) Show that β is the function of K, q, α
given by solving the equation
β = 1 − (1 − α) exp (−Kqβ) .
Notice that β is NOT equal to 1, except in the trivial case α = 1. That is, no matter how long the
epidemic goes on, some plants will miraculously stay healthy. Draw a graph of β as a function of
q (measured in weeks), if K = 0.3 per week and α = 0.3.
7. Staying with potato blight: using the notation in the lecture notes (and in Question 6), explain
how you would find x(t) for t ∈ [2p, 2p + q]. You can just exhibit the differential equation AND
its “initial condition” (that is, tell me x(2p)); you don’t need to solve it unless you feel the need
for some exercise.
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