unit #16 : di erential equations goals: to develop skills …math121/notes/notes16.pdf ·...

Unit #16 : Differential Equations

Goals:

• To develop skills needed to find the appropriate differential equations to use asmathematical models.

Differential Equation Modelling - 1

Differential Equation Modelling

In working with the models in in this topic, it should not be your goal tomemorize particular differential equations and their solutions as ifthey were the correct formulas for the problems under discussion.Instead, you should concentrate on the process that leads to the differential equa-tions, and the general solution approaches once you have the equation.You should try to become comfortable with the various parts of this process, whichinvolves:

• translating relationships derived from common sense or elementary science, ex-pressed in ordinary English, into mathematical equations,

• judgments (regarding which effects are insignificant enough to disregard orwhich factors vary little enough to be regarded as constant), and

• recognizing special cases that are amenable to solution.

Differential Equation Modelling - 2

DE Models of Growth and Decay

We have already seen that exponential growth and decay have simple formulationsusing differential equations. A summary is shown in the box below.

Differential Equation for Exponential Growth/Decay

Every solution to the equation

dP

dt= k P

can be written in the form

P (t) = P0 ekt

where P0 is the initial value of P ;

Equations in which k > 0 represent growing systems and with k < 0 representsdecaying ones.

We now proceed to study some related modeling problems using differential equa-tions.

Newton’s Law of Heating and Cooling - 1

Example (Newton’s Law of Heating and Cooling)Newton’s Law of Cooling asserts that the rate at which an object cools off (orheats up) is proportional to the difference between the temperature of the objectand that of its environment.

dH

dt= −α (H − Tenv)

where

• H is the temperature of the object,

• Tenv is the temperature of the environment, and

• α is a property of the object, involving such things as the ability of the surfaceof the object to conduct heat.

Explain why α must be positive.


Example: If we put a potato at room temperature (20 ◦C) in an oven at200 ◦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 ofthe potato.


dH

dt= −0.25(H − 200)

Solve this differential equation to get an equation for the temperature of thepotato, and use the equation to predict the potato’s temperature after 10 min-utes.


dH

dt= −0.25(H − 200)


Sketch a slope field for the DE

dH

dt= −0.25 (H − 200)

and draw in some solutions, including one for which the initial condition isgreater than 200 ◦C.

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16 18 20

t (min)

H(t) (o C)

Equilibrium Solutions - 1

Equilibrium SolutionsIt is important to realize that although solving differential equations can be hard(because solving implicitly involves integration), we can learn important character-istics about our model directly from the differential equation itself. One importantfeature of many models are the equilibrium solution(s).

For the potato problem, the DE we found wasdH

dt= −0.25 (H − 200). By

considering only this equation, what would be the equilibrium temperature(s)of the potato.

(a) H = 0

(b) H = 100

(c) H = 200

(d) H = −0.25(200)


Example: For water leaking out of a cylindrical reservoir under the force ofgravity, we can use conservation of energy arguments to derive the differentialequation for the depth of the water, h:

dh

dt= −k

√h.

What is the equilibrium level of water?

(a) h = 0

(b) h = −k

(c) h = 1

(d) h = 10


Example: For the logistic population model,

dP

dt= −kP (L− P ),

what is/are the equilibrium populations?

(a) P = 0

(b) P =L

2

(c) P = 0, L

(d) P = 0,L

2, L

Modelling - Ice Formation - 1

Applications and Modeling

As the textbook indicates, a mathematical model is sometimes obtained by findinga function that fits certain experimental measurements, or by a theoretical approachthat leads directly to a certain formula, or by a combination of the two approaches.The models that follow are chosen to illustrate this idea.


Example: Thickness of IceThe surface of water freezes first, through contact with cold air. As heat fromthe 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 icebegins to form. The rate at which ice forms is proportional to the rate atwhich heat escapes from the water to the air. The rate at which heat escapesfrom the water (through the ice) to the air is inversely proportional to thethickness of the ice.

Translate the rate information into a differential equation, and an initial con-dition.


Solve this differential equation, subject to the initial condition.


If it takes two hours to form 1/2 cm of ice, then how long will it take to form2 cm?

Modelling - Drug Dosing - 1

Example: Drug DosingConsider a patient who is being given morphine at a rate of 2.5 mg/hr. Their bodymetabolizes the drug at a continuous rate of 30% per hour.Write a differential equation for the amount of morphine in the patient overtime.


Find the general solution to the differential equation.


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


Based on the earlier analysis, sketch several solutions to morphine DE on theaxes below.

Modelling - Terminal Velocity - 1

Example: Terminal VelocityIn physics, we note that an object near the earth will accelerate at 9.8 m/s2. Thiswould imply that, if we drop an object from high enough, it will accelerate until itreaches 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 takeabout 8 seconds to reach the ground, and have the object falling at 78 m/s or 280km/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 toreach the ground, and would be falling at 580 m/s or almost twice the speed ofsound.


Why don’t we observe objects falling this fast in practice?

Draw a diagram indicating the forces on a falling object.


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.


Explain what happens to an object that is traveling slower than the equilibriumvelocity.

Explain what happens to an object that is traveling faster than the equilibriumvelocity.


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


Sketch the shape of the possible graphs of v(t) that satisfy the DE.

Modelling - Logistic Growth - 1

Example: Logistic GrowthThe logistic model is a model for population growth that takes into account thefact that population growth slows down when the environment occupied by thepopulation is saturated. In many cases, the environment has a certain carryingcapacity L which represents a degree of overpopulation that would prevent furthergrowth. In the logistic model, it is assumed that the rate of growth of the populationP (t) is proportional to the product of P (t) and (L− P (t)).

Write down the differential equation that expresses this relationship.


Recall that when we studied this differential equation in an earlier unit, we foundthat the growth rate of the population is largest when the size of the population ishalfway between 0 (no population) and L (max sustainable population).

When the population size, P , is close to 0 or close to L, we found the rate ofgrowth to be very small. When the population size, P , is greater than L, the rateof growth of P becomes negative.Find the equilibrium levels of the population.


Show that the population will always increase if the initial population is in therange 0 < P < L.

Show that the population will always decrease if the initial population is aboveL.

Modelling - Compartmental Analysis - 1

Compartmental Analysis I

An aquarium holding 25 m3 of water at a zoo is kept clean using a large pipesystem equipped with filters. Water is pumped into and out of the aquarium atconstant rate of 10 m3 per day. In the pipe flowing into the aquarium, a filter hasbroken down, so ammonia (toxic to fish) is flowing into the tank with the water,at a concentration of 150 g / m3. The water in the aquarium starts out at timet = 0 free of ammonia.Find the rate of ammonia entering the aquarium in grams/day.


If the water in the aquarium is well mixed (same concentration of ammoniaeverywhere within the tank), and the water is leaving the tank at 10 m3/day,find an expression for the rate that ammonia is leaving the tank, given thatthe amount of ammonia in the tank at time t is S(t) grams.


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 concen-

tration of ammonia,dC

dt.

Compartmental Analysis II - 1

Compartmental Analysis II

Use the differential equation to find the equilibrium concentration of ammo-nia in the aquarium. Explain why the equilibrium level makes sense for the

situation.dC

dt= 60− C

2.5


Find C(t), the solution to the differential equation, given that the initial am-

monia concentration is zero.dC

dt= 60− C

2.5


dC

dt= 60− C

2.5


A concentration of 0.1 mg/l (or 100 g/m3) is considered dangerous for thefish in the tank. How long after the filter breakdown do the fish have for theproblem to be identified and fixed before the ammonia levels become danger-ous?

unit #16 : di erential equations goals: to develop skills …math121/notes/notes16.pdf ·...

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