Goals: To develop skills needed to find the appropriate differential equations to use as mathematical models.

Transcription

1 Unit #17 : Differential Equations Goals: To develop skills needed to find the appropriate differential equations to use as mathematical models. Reading: Sections In workingwiththemodels insections , it should not be your goal to memorize particular differential equations and their solutions as if they were the correct formulas for the problems under discussion. Instead, you should concentrate on the process that leads to the differential equations.

2 2 You should try to become comfortable with the various parts of this process, which involves translating relationships derived from common sense or elementary science, expressed in ordinary English, into mathematical equations, judgments (regarding which effects are insignificant enough to disregard or which factors vary little enough to be regarded as constant), and recognizing special cases that are amenable to solution.

3 Unit 17 Differential Equations 3 DE Models of Growth and Decay Reading: Section 11.5 in the textbook. We have already seen that exponential growth and decay have simple formulations using differential equations. A summary is shown in the box below. Differential Equation for Exponential Growth/Decay Every solution to the equation dp dt = kp can be written in the form P(t) = P 0 e kt where P 0 is the initial value of P; Equations in which k > 0 represent growing systems and with k < 0 represents decaying ones. We now proceed to study some related modeling problems using differential equations.

4 4 Example (Newton s Law of Cooling) Newton s Law of Cooling asserts that the rate at which an object cools off (or heats up) is proportional to the difference between the temperature of the object and that of its environment. where H is the temperature of the object, dh dt = α(h T env), Tenv is the temperature of the environment, and α is a property of the object, involving such things as the ability of the surface of the object to conduct heat. Explain why α must be positive.

5 Unit 17 Differential Equations 5 Example: If we put a potato at room temperature (20 C) in an oven at 200 C, then Tenv = 200. Suppose that α = 0.25 and t is measured in minutes. Write the differential equation describing the rate of temperature change of the potato.

6 6 Solve this differential equation to get an equation for the temperature of the potato, and use the equation to predict the potato s temperature after 10 minutes.

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8 Unit 17 Differential Equations 7 Sketch a slope field for the DE dh = 0.25(H 200) dt and draw in some solutions, including one for which the initial condition is greater than 200 C.

9 H (deg. C) t (mins)

10 Unit 17 Differential Equations 9 Equilibrium Solutions It is important to realize that although solving differential equations can be hard (because solving implicitly involves integration), we can learn important characteristics about our model directly from the differential equation itself. One important feature of many models are the equilibrium solution(s). For the potato problem, the DE we found was dh = 0.25(H 200). By considering only this equation, find the equilibrium temperature(s) of the dt potato.

11 10 For water leaking out of a cylindrical reservoir under the force of gravity, we can use conservation of energy arguments to derive the differential equation dh for the depth of the water, h: dt = k h. Find the equilibrium level of water. [See HH pg for further equilibrium discussion.]

12 Unit 17 Differential Equations 11 Applications and Modeling Reading: 11.6 and 11.7 in the textbook. As the textbook indicates, a mathematical model is sometimes obtained by finding a function that fits certain experimental measurements, or by a theoretical approach thatleads directlyto a certain formula, orbyacombinationof the twoapproaches. The models in the book are chosen to illustrate this idea. Example: Thickness of Ice The surface of water freezes first, through contact with cold air. As heat from the water travels up through the ice and is lost to the air, more ice is formed. We want a function describing the thickness of ice as a function of time. Let y(t) = the thickness at time t, measured from the time at which the ice begins to form. The rate at which ice forms is proportional to the rate at which heat escapes from the water to the air. The rate at which heat escapes from the water (through the ice) to the air is inversely proportional to the thickness of the ice.

13 12 Translate the rate information into a differential equation, and an initial condition.

14 Unit 17 Differential Equations 13 Solve this differential equation, subject to the initial condition.

15 14 If it takes two hours to form 1/2 cm of ice, then how long will it take to form 2 cm?

16 Unit 17 Differential Equations 15 Example: Drug Dosing Considerapatientwho is being given morphine at a rate of 2.5 mg/hr. Their body metabolizes the drug at a rate of 30% per hour. Write a differential equation for the amount of morphine in the patient over time.

17 16 In the long run, how much morphine will the patient have in their body?

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19 Unit 17 Differential Equations 17 Based on the earlier analysis, sketch several solutions to morphine DE on the axes below.

20 18 Example: Terminal Velocity In physics, we note that an object near the earth will accelerate at 9.8 m/s 2. This would imply that, if we drop an object from high enough, it will accelerate until it reaches any velocity we want. For example, the main visitor s gallery of the CN tower is 346 m above the ground. Using just the force of gravity, an object dropped from this height should take about 8 seconds to reach the ground, and have the object falling at 78 m/s or 280 km/h, or 1/4 the speed of sound. If you dropped an object out of a plane at 12,000 m, it would take 50 seconds to reach the ground, and would be falling at 580 m/s or almost twice the speed of sound.

21 Unit 17 Differential Equations 19 Why don t we observe objects falling this fast in practice? Draw a diagram indicating the forces on a falling object.

22 20 Assuming that the force exerted by air resistance is proportional to velocity, write a differential equation for the velocity of a falling object. Find the equilibrium velocity based on the differential equation.

23 Unit 17 Differential Equations 21 Explain what happens to an object that is traveling slower than the equilibrium velocity. Explain what happens to an object that is traveling faster than the equilibrium velocity.

24 22 Solve the differential equation to obtain an expression for v(t).

25 Unit 17 Differential Equations 23 Sketch the shape of the possible graphs of v(t) that satisfy the DE.

26 24 Example: Logistic Growth The logistic model is a model for population growth that takes into account the fact that population growth slows down when the environment occupied by the population is saturated. In many cases, the environment has a certain carrying capacity L which represents a degree of overpopulation that would prevent further growth. Inthelogisticmodel,itisassumedthattherateofgrowthofthepopulation P(t) is proportional to the product of P(t) and L P(t). Write down the differential equation that expresses this relationship.

27 Unit 17 Differential Equations 25 Recall that when we studied this differential equation in Unit #16, we found that the growth rate of the population is largest when the size of the population is halfway between 0 and L. When the population size, P, is close to 0 or close to L, we found the rate of growth to be very small. When the population size, P, is greater than L, the rate of growth of P becomes negative. Find the equilibrium levels of the population.

28 26 Show that the population will always increase if the initial population is in the range 0 < P < L. Show that the population will always decrease if the initial population is below L.

29 Unit 17 Differential Equations 27 Example (Pendulum) A very simple analysis of the forces acting on a pendulum of length l with a mass m leads to the following differential equation for the dependence on time of the angle θ between the pendulum and a vertical line through the pivot point P: d 2 θ dt 2 = g L sinθ. This differential equation cannot be solved by any exact method. You do not need to understand the physics to do the following question.

30 28 If the amplitude of the pendulum s swing is small, then θ stays close to zero. What function is approximately equal to sinθ for values of θ close to zero? (You should use linear approximations to answer this). Rewrite the DE with sinθ replaced by the function you found, and use your knowledge of derivatives to guess a form of the solution for the differential equation.

31 Unit 17 Differential Equations 29 Compartmental Analysis An aquarium holding 25 m 3 of water at a zoo is kept clean using a large pipe system equipped with filters. Water is pumped into and out of the aquarium at constant rate of 10 m 3 per day. In the pipe flowing into the aquarium, a filter has broken down, so ammonia (toxic to fish) is flowing into the tank with the water, at a concentration of 150 g / m 3. The water in the aquarium starts out at time t = 0 free of ammonia. Find the rate of ammonia entering the aquarium in grams/day.

32 30 If the water in the aquarium is well mixed (same concentration of ammonia everywhere within the tank), and the water is leaving the tank at 10 m 3 /day, find an expression for the rate that ammonia is leaving the tank, given that the amount of ammonia in the tank at time t is S(t) grams.

33 Unit 17 Differential Equations 31 Use your two rates to find an expression for the net rate of change of the ammonia mass, ds dt. Convert your differential equation to show the rate of change of the concentration of ammonia, dc dt.

34 32 Use the differential equation to find the equilibrium concentration of ammonia in the aquarium. Explain why the equilibrium level makes sense for the situation.

35 Unit 17 Differential Equations 33 Find C(t), the solution to the differential equation, given that the initial ammonia concentration is zero.

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37 34 A concentration of 0.1 mg/l (or 100 g/m 3 ) is considered dangerous for the fish in the tank. How long after the filter breakdown do the fish have for the filter problem to be identified and fixed before the ammonia levels become dangerous?