MATH1400: Modelling with differential equations, 2011–12Examples 3 Professor A.M. Rucklidge, 8.17f, Department of Applied Mathematics. Course web page: http://www.maths.leeds.ac.uk/~alastair/MATH1400/. Please do section 1 during the tutorials on Fridays 2 and 9 March. There will also be a quiz on Friday 2 March. The topics for the quiz will be announced during lectures. Hand in your answers to section 2 in the grey boxes outside the School of Maths Under- graduate Office by 5pm on Monday 12th March. Be sure to put your work in your tutor’s box. Be sure to write your name on your work. Your work will be returned during tutorials or in the blue trays outside the School of Maths Undergraduate Office. In all cases, clearly specify the ODE that must be solved, along with its initial condition, and use the appropriate method to find the particular solution. Where the question asks for an interpretation, be sure to give one that is independent of the choice of name or coordinate system you used to solve the problem. Section 1: to be covered in tutorials. 1. A nuclear breeder reactor produces waste that contains (amongst other things) the isotope 239 Pu (Plutonium-239). After 15 years, the initial concentration of 239 Pu in the waste has decreased by 0.043%. Find the half-life of the isotope. 2. A particle is acted on by an oscillating force; there is also an air resistance force that opposes the motion. The speed v of the particle is governed by the ODE dv = F sin t − kv, dt where F is proportional to the strength of the oscillating force and k is a constant giving the size of the air resistance. What kind of ODE is this? Suppose that at time t = 0, the speed is v0 . Find v(t). 3. Solid cancerous tumours do not grow exponentially in time. As the tumour becomes larger, the doubling time increases. Research has shown that the number of cells n(t) in a tumour at time t satisfies the equation λ −αt n(t) = n0 exp 1−e , α where n0 , λ and α are constants. (a) Show that n satisfies the differential equation dn = λe−αt n, with n(0) = n0 . dt (b) A tumour satisfies the above equation with α = 0.02. Originally, when it contained 104 cells, the tumour was increasing at a rate of 20% per unit time. What is the limiting number of cells in the tumour? 1 8x − 0. Estimate when the murder occurred. the pollutant concentration of Lake Erie (caused by past industrial pollution) is five times that of Lake Huron. Show that we can model the amount of pollution x(t) in Lake Erie by dx r o = ri ci − x dt V How long will it take to reduce the pollution concentration in Lake Erie to the level of twice that of Lake Huron? Section 2: to be handed in 1. Find the general solution p(t). the temperature of the body was 31◦ C. Assume that the outflow from Lake Erie is perfectly mixed lake water. As the salt KNO3 dissolves in methanol.3% of the original 14 C remained.m. find the particular solution. Suppose that at the time t = 0 (years).) 2. By 8 a.04p 1 − 6 where t is measured in years. It was found that 92. (In 2005. with Θ(0) = Θ0 . dt Solve this equation. 4. and that the concentration of pollution (ci ) in Lake Huron does not change. 5. the number x(t) of grams of the salt in a solution after t seconds satisfies the differential equation dx = 0.004x2 . A body is found at the scene of a crime at 6 a. A small piece of the Turin Shroud was subjected to Carbon dating in 1988. it was suggested that the 1988 sample was taken from an area where the original shroud might have been repaired. and normal body temperature is 37◦ C. The population p(t) of the county of Midsommershire obeys the Logistic equation: dp p = 0. – at that time. The ambient temperature was 20◦ C.m. Suppose that the ambient temperature varies sinusoidally in time. and describe the solution in the limit t → ∞. how long will it take for an additional 50g of salt to dissolve? 2 . What happens as t → ∞? 3. the temperature had cooled to 29◦ C.. (b) If the population was 8×106 in 1970. 4. dt 10 (a) Modify the equation to take into account the fact that 10 000 people move away from Midsommershire each year. dt (a) What is the maximum amount of the salt that will ever dissolve in the methanol? (b) If x = 50 when t = 0. Then Newton’s Law of cooling becomes: dΘ = −k(Θ − A sin(ωt)). Calculate (to the nearest year) the year that this suggests the shroud was made. Assume that Lake Erie has a volume V of 480 km3 of water and that its rate of inflow ri (from Lake Huron) and outflow ro (to Lake Ontario) are both 350 km3 per year. You may leave your answer in terms of logs. and that draws out stale air at the same rate. It is ventilated by a system that pumps in fresh air at a rate r. or use a calculator. A lecture theatre contains a volume of air V = 120 000 m3 . The ventilation system is required to be able to reduce any impurities to a level of 1% of their initial concentration in 30 min.5. 3 . Find the required pumping rate r (in m3 /min). Assume that the air in the theatre is mixed well.