[SOLVED] One or more initial conditions are given

> In the study of the effect of natural selection on a population, we encounter the differential equation dq/dt = -.0001q2 (1 - q), where q is the frequency of a gene a and the selection pressure is against the recessive genotype aa. Sketch a solution of

> Show that the mathematical model in Check Your Understanding 2 predicts that the amount of litter in the forest will eventually stabilize. What is the “equilibrium level” of litter in that problem? [Note: Today, most forests are close to their equilibriu

> A single dose of iodine is injected intravenously into a patient. The iodine mixes thoroughly in the blood before any is lost as a result of metabolic processes (ignore the time required for this mixing process). Iodine will leave the blood and enter the

> On the slope field in Fig. 4(a) or a copy of it, draw an approximation of a portion of the solution curve of the differential equation y’ = t - y that goes through the point (0, 2). In your opinion, based on the slope field, can this solution pass throu

> A certain drug is administered intravenously to a patient at the continuous rate of 5 milligrams per hour. The patient’s body removes the drug from the bloodstream at a rate proportional to the amount of the drug in the blood. Write a differential equati

> The air in a crowded room full of people contains .25% carbon dioxide (CO2). An air conditioner is turned on that blows fresh air into the room at the rate of 500 cubic feet per minute. The fresh air mixes with the stale air, and the mixture leaves the r

> A company arranges to make continuous deposits into a savings account at the rate of P dollars per year. The savings account earns 5% interest compounded continuously. Find the approximate value of P that will make the savings account balance amount to $

> A company wishes to set aside funds for future expansion and so arranges to make continuous deposits into a savings account at the rate of $10,000 per year. The savings account earns 5% interest compounded continuously. (a) Set up the differential equat

> A continuous annuity is a steady stream of money that is paid to some person. Such an annuity may be established, for example, by making an initial deposit in a savings account and then making steady withdrawals to pay the continuous annuity. Suppose tha

> The fish population in a pond with carrying capacity 1000 is modeled by the logistic equation dN/dt = .4 1000 N(1000 - N ). Here, N(t) denotes the number of fish at time t in years. When the number of fish reached 275, the owner of the pond decided t

> Consider a certain commodity that is produced by many companies and purchased by many other firms. Over a relatively short period, there tends to be an equilibrium price p0 per unit of the commodity that balances the supply and the demand. Suppose that,

> In economic theory, the following model is used to describe a possible capital investment policy. Let f (t) represent the total invested capital of a company at time t. Additional capital is invested whenever f (t) is below a certain equilibrium value E,

> L. F. Richardson proposed the following model to describe the spread of war fever. If y = f (t) is the percentage of the population advocating war at time t, the rate of change of f (t) at any time is proportional to the product of the percentage of the

> Suppose that substance A is converted into substance B at a rate that, at any time t, is proportional to the square of the amount of A. This situation occurs, for instance, when it is necessary for two molecules of A to collide to create one molecule of

> The health officials that studied the flu epidemic in Example 8 made an error in counting the initial number of infected people. They are now claiming that f (t) (the number of infected people after t days) is a solution of the initial-value problem y’

> An experimenter reports that a certain strain of bacteria grows at a rate proportional to the square of the size of the population. Set up a differential equation that describes the growth of the population. Sketch a solution.

> Let c be the concentration of a solute outside a cell that we assume to be constant throughout the process, that is, unaffected by the small influx of the solute across the membrane due to a difference in concentration. The rate of change of the concentr

> A porous material dries outdoors at a rate that is proportional to the moisture content. Set up the differential equation whose solution is y = f (t), the amount of water at time t in a towel on a clothesline. Sketch the solution.

> In an autocatalytic reaction, one substance is converted into a second substance in such a way that the second substance catalyzes its own formation. This is the process by which trypsinogen is converted into the enzyme trypsin. The reaction starts only

> At one point in his study of a falling body starting from rest, Galileo conjectured that its velocity at any time is proportional to the distance it has dropped. Using this hypothesis, set up the differential equation whose solution is y = f (t), the dis

> For information being spread by mass media, rather than through individual contact, the rate of spread of the information at any time is proportional to the percentage of the population not having the information at that time. Give the differential equat

> Answer parts (a) and (b) in Example 2 if the pond has a carrying capacity of 2000 fish and all other data are unchanged. Example 2: A pond on a fish farm has a carrying capacity of 1000 fish. The pond was originally stocked with 100 fish. Let N(t) denot

> Answer part (a) in Example 2 if the pond was originally stocked with 600 fish and all other data are unchanged, how does the graph of the fish population in this case differ from the one in Example 2? Example 2: A pond on a fish farm has a carrying capa

> You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of dN/dt versus N in an Nz-plane. (c) In the tN-plane, plot the constant solutions and place a dashed l

> On the slope field in Fig. 5(a) or a copy of it, draw the solution of the initial-value problem yâ = .0002y (5000 - y), y(0) = 500. Figure 5: 5000 4000 3000 2000 1000 0 0 1 2 3 4 5 6 t (a) Slope field of y = .0002y(5000 - y). y In

> Sketch the graph of a function with the stated properties. Domain: 0 ≤ t ≤ 4; (0, 2) is on the graph; the slope is always positive, and the slope becomes more positive (as t increases).

> Sketch the graph of a function with the stated properties. Domain: 0 ≤ t ≤ 3; (0, 1) is on the graph; the slope is always positive, and the slope becomes less positive (as t increases).

> Draw the graph of g(x) = ex - 100x2 - 1, and use the graph to sketch the solution of the differential equation y’ = ey - 100y2 - 1 with initial condition y(0) = 4 on a ty-coordinate system.

> Draw the graph of g(x) = (x - 2)2 (x - 6)2, and use the graph to sketch the solutions of the differential equation y’ = (y - 2)2 (y - 6)2 with initial conditions y(0) = 1, y(0) = 3, y(0) = 5, and y(0) = 7 on a ty-coordinate system.

> A parachutist has a terminal velocity of -176 feet per second. That is, no matter how long a person falls, his or her speed will not exceed 176 feet per second, but it will get arbitrarily close to that value. The velocity in feet per second, y(t), after

> Suppose that, once a sunflower plant has started growing, the rate of growth at any time is proportional to the product of its height and the difference between its height at maturity and its current height. Give a differential equation that is satisfied

> One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not alre

> Verify that the function f (t) = 2e-t + t - 1 is a solution of the initial-value problem yâ = t - y, y(0) = 1. [This is the function shown in Fig. 4(c). In Section 10.3, you will learn how to derive this solution.] Figure 4: M co

> One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not alre

> The slope field in Fig. 4(a) suggests that the solution curve of the differential equation yâ = t - y through the point (0, -1) is a straight line. (a) Assuming that this is true, find the equation of the line. (b) Verify that the funct

> One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not alre

> When the breath is held, carbon dioxide (CO2) diffuses from the blood into the lungs at a steadily decreasing rate. Let P0 and Pb denote the pressure of CO2 in the lungs, respectively, in the blood at the moment when the breath is held. Suppose that Pb i

> One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not alre

> Sketch the graph of a function with the stated properties. Domain: 0 ≤ t ≤ 8; (0, 6) is on the graph; the slope is always negative, the slope becomes more negative as t increases from 0 to 3, and the slope becomes less negative as t increases from 3 to 8

> Sketch the graph of a function with the stated properties. Domain: 0 ≤ t ≤ 7; (0, 2) is on the graph; the slope is always positive, the slope becomes more positive as t increases from 0 to 3, and the slope becomes less positive as t increases from 3 to 7

> Sketch the graph of a function with the stated properties. Domain: 0 ≤ t ≤ 6; (0, 4) is on the graph; the slope is always negative, and the slope becomes more negative.

> Sketch the graph of a function with the stated properties. Domain: 0 ≤ t ≤ 5; (0, 3) is on the graph; the slope is always negative, and the slope becomes less negative.

> A cool object is placed in a room that is maintained at a constant temperature of 20C. The rate at which the temperature of the object rises is proportional to the difference between the room temperature and the temperature of the object. Let y = f (t) b

> A person deposits $10,000 in a bank account and decides to make additional deposits at the rate of A dollars per year. The bank compounds interest continuously at the annual rate of 6%, and the deposits are made continuously into the account. (a) Set up

> A person planning for her retirement arranges to make continuous deposits into a savings account at the rate of $3600 per year. The savings account earns 5% interest compounded continuously. (a) Set up a differential equation that is satisfied by f (t),

> Refer to Example 2. Answer questions (a) and (b) in Exercise 1 if the interest rate is 7%. How long will it take to pay off the $25,000 loan in this case? Example 2: You took a loan of $25,000 to pay for a new car. The interest rate on the loan is 5%. Y

> Refer to Example 1. (a) How fast was the amount in the account growing when it reached $30,000? (b) How much money was in the account when it was growing at twice the rate of your annual contribution? (c) How long do you have to wait for the money in the

> A certain drug is administered intravenously to a patient at the continuous rate of r milligrams per hour. The patient’s body removes the drug from the bloodstream at a rate proportional to the amount of the drug in the blood, with constant of proportion

> Morphine is a drug that is widely used for pain management. However, morphine can cause fatal respiratory arrest. Since pain perception and drug tolerance vary with patients, morphine is gradually administered with small increments until pain is controll

> Figure 4 contains the solution of the initial-value problem in Exercise 23. Figure 4: (a) With the help of the graph, approximate how long it will take before the account is depleted. (b) Solve the initial-value problem to determine P(t). (c) Use the

> After depositing an initial amount of $10,000 in a savings account that earns 4% interest compounded continuously, a person continued to make deposits for a certain period of time and then started to make withdrawals from the account. The annual rate of

> You make an initial deposit of $500 in a savings account and plan on making future deposits at a gradually increasing annual rate given by 90t + 810 dollars per year, t years after the initial deposit. Assume that the deposits are made continuously and t

> A person deposits an inheritance of $100,000 in a savings account that earns 4% interest compounded continuously. This person intends to make withdrawals that will increase gradually in size with time. Suppose that the annual rate of withdrawals is 2000

> Let f (t) denote the amount of capital invested by a certain business firm at time t. The rate of change of invested capital, f ‘(t), is sometimes called the rate of net investment. The management of the firm decides that the optimum level of investment

> Radium 226 is a radioactive substance with a decay constant .00043. Suppose that radium 226 is being continuously added to an initially empty container at a constant rate of 3 milligrams per year. Let P(t) denote the number of grams of radium 226 remaini

> According to the National Kidney Foundation, in 1997 more than 260,000 Americans suffered from chronic kidney failure and needed an artificial kidney (dialysis) to stay alive. (Source: The National Kidney Foundation, www.kidney.org.) When the kidneys fai

> Find a formula for P(t) in Exercise 17 if, initially, 10,000 bacteria were present in the culture. Exercise 17: In an experiment, a certain type of bacteria was being added to a culture at the rate of e0.03t + 2 thousand bacteria per hour. Suppose that

> In an experiment, a certain type of bacteria was being added to a culture at the rate of e0.03t + 2 thousand bacteria per hour. Suppose that the bacteria grow at a rate proportional to the size of the culture at time t, with constant of proportionality k

> Derive the formula for the population in Example 3, if the population in 1995 was 2 million. (The formula is given following the solution of Example 3.) Example 3: In 2005, people in a country suffering from economic problems started to emigrate to othe

> A body was found in a room when the room’s temperature was 70˚F. Let f (t) denote the temperature of the body t hours from the time of death. According to Newton’s law of cooling, f satisfies a differential equation of the form y = k(T - y). (a) Find

> Rework Exercise 13 for a metal with a constant of proportionality k = .2. Which rod cools faster, the rod with a constant of proportionality k = .1 or the rod with a constant of proportionality k = .2? What can you say about the effect of varying the con

> When a red-hot steel rod is plunged in a bath of water that is kept at a constant temperature 10˚C, the temperature of the rod at time t, f (t), satisfies the differential equation y = k[10 - y], where k > 0 is a constant of proportionality. Determin

> Find the demand function if the elasticity of demand is a linear function of price given by E( p) = ap + b, where a and b are constants.

> Let q = f (p) be the demand function for a certain commodity, where q is the demand quantity and p the price of 1 unit. In Section 5.3, we defined the elasticity of demand as E( p) = -p f ‘( p) / f (p). (a) Find a differential equation satisfied by th

> Let f (t) be the size of a paramecium population after t days. Suppose that y = f (t) satisfies the differential equation y’ = .003y (500 - y), y (0) = 20. Describe this initial-value problem in words.

> Find a constant solution of y’ = t2y - 5t2.

> Play Time Toys is organised into two major divisions: marketing and production. The production division is further divided into three departments: puzzles, dolls and video games. Each production department has its own manager. The companyâ&#1

> Baker Street Animal Clinic uses a particular serum routinely in its vaccination program. Veterinarian technicians give the injections. The standard dose is 10cc per injection, and the cost has been $100 per 1000cc. According to records, 2000 injections w

> The Mighty Morphs produces two popular games, Powerful Puffs and Mini-Mite Morphs. Following are standard costs: The standards call for more than one disk and documentation book per unit because of normal waste due to faulty DVDs and poor binding. Act

> Palm Producers (PP) is expecting sales growth, and so it built nearly identical automated plants in Sandy Beach, Queensland, and in Singapore to produce its new Palm Powerhouse. Each plant manager is responsible for producing adequate inventories to meet

> Your brother started a small business, GameZ, that produces a software game he developed. It is his first year in business, and he kept detailed records of the business. However, his business records consist primarily of entries in his chequebook plus in

> Northcoast Manufacturing Company, a small manufacturer of parts used in appliances, just completed its first year of operations. The companyâs controller, Vic Trainor, has been reviewing the actual results for the year and is concerned

> Security Vehicles converts Hummers into luxury, high-security vehicles by adding a computerised alarm and radar system and various luxury components. The finished vehicles are sold for $100 000 each. Variable manufacturing costs (including the cost of th

> Fighting Kites produces several different kite kits. Last year, the company produced 20Â 000 kits and sold all but 2000 kits. The kits sell for $30 each. Costs incurred are listed here. Beginning inventory last year held 2000 kits. Assume th

Question: One or more initial conditions are given