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