Assignment 4 OMAS120 Applied Mathematics Study Period 3, 2014

Assignment 4
Due : 4pm Friday November 21
OMAS120 Applied Mathematics
Study Period 3, 2014
Mathematics and Statistics
1. Solve the following differential equations subject to the stated condition on y(x)
dy
y
= √ with y(0) = 1.
dx
x
dy
(ii) x
− 3y + 2 = 0 with y(1) = 1
dx
(i)
2. A polluted river with a nutrient concentration of 90g/m3 is flowing at a rate of 100m3 /day
into an estuary of volume 1000m3 . At the same time, water from the estuary is flowing
into the ocean at 100m3 /day. The initial nutrient concentration in the estuary is 20g/m3 .
(i) Let N (t) be the amount of nutrient (in grams) in the estuary at time t. Write down
and solve an appropriate differential equation for N (t) along with the appropriate
initial condition.
(ii) After a long time, what is the concentration of nutrient in the estuary?
(iii) It is known that if the nutrient concentration in the estuary reaches 70g/m3 an algal
bloom will occur. How many days does it take for the nutrient concentration to reach
this threshold?
3. Find the derivative for each of the following functions.
(i) f (x) = e2x sin2 x
(ii) f (x) = ln |2 cos x|
(iii) f (x) = tan−1 (x2 + 1) (Note: tan−1 is the alternative notation for arctan.)
4. Find the following integrals.
Z 1
(i)
x2 sin x3 dx
0
Z
(ii)
x3 (1 + x8 )−1 dx
5. The suspension in a car acts like a damped harmonic oscillator, that is, the oscillations in
the suspension rapidly die down with time. A model for this includes both exponential
and trigonometric functions. Suppose the displacement in a car’s suspension is given by
s(t) = e−t/2 cos(2t)
(i) Sketch the displacement of the suspension for 0 ≤ t ≤ 2π and describe its behaviour
in a few words.
(ii) Show by direct substitution that the displacement satisfies the differential equation
4
d2 s
ds
+ 4 + 17s = 0.
2
dt
dt
Notes:
• 10% of the marks for this assignment are reserved for presentation.
• There are penalties for late assignments. You must contact your tutor before the due date
if you have difficulties making the deadline.
1