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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)

(i) dy

dx =

y √ x

with y(0) = 1.

(ii) x dy

dx − 3y + 2 = 0 with y(1) = 1

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.

(i)

∫ 1 0 x2 sin x3 dx

(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 d2s

dt2 + 4

ds

dt + 17s = 0.

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.

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