Differential Equations with Boundary-Value Problems, 8th Edition solutions

Differential Equations with Boundary-Value Problems, 8
The population of a community is known to increase at a rate proportional to the number of people present at time t. If an initial population P0 has doubled in 5 years, how long will it take to triple? To quadruple?

Differential Equations with Boundary-Value Problems, 8
Suppose it is known that the population of the community in Problem 1 is 10,000 after 3 years. What was the initial population P0? What will be the population in 10 years? How fast is the population growing at t 10?

Differential Equations with Boundary-Value Problems, 8
The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 15% in 10 years. What will be the population in 30 years? How fast is the population growing at t 30?

Differential Equations with Boundary-Value Problems, 8
The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. After 3 hours it is observed that 400 bacteria are present. After 10 hours 2000 bacteria are present. What was the initial number of bacteria?

Differential Equations with Boundary-Value Problems, 8
The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t and has a halflife of 3.3 hours. If 1 gram of this isotope is present initially, how long will it take for 90% of the lead to decay?

Differential Equations with Boundary-Value Problems, 8
Initially 100 milligrams of a radioactive substance was present. After 6 hours the mass had decreased by 3%. If the rate of decay is proportional to the amount of the substance present at time t, find the amount remaining after 24 hours.

Differential Equations with Boundary-Value Problems, 8
Determine the half-life of the radioactive substance described in Problem 6.

Differential Equations with Boundary-Value Problems, 8
(a) Consider the initial-value problem dAdt kA, A(0) A0 as the model for the decay of a radioactive substance. Show that, in general, the half-life T of the substance is T (ln 2)k. (b) Show that the solution of the initial-value problem in part (a) can be written A(t) A02t/T. (c) If a radioactive s

Differential Equations with Boundary-Value Problems, 8
When a vertical beam of light passes through a transparent medium, the rate at which its intensity I decreases is proportional to I(t), where t represents the thickness of the medium (in feet). In clear seawater, the intensity 3 feet below the surface is 25% of the initial intensity I0 of the inciden

Differential Equations with Boundary-Value Problems, 8
When interest is compounded continuously, the amount of money increases at a rate proportional to the amoun S present at time t, that is, dSdt rS, where r is the annual rate of interest. (a) Find the amount of money accrued at the end of 5 years when $5000 is deposited in a savings account drawing 5

Differential Equations with Boundary-Value Problems, 8
Archaeologists used pieces of burned wood, or charcoal, found at the site to date prehistoric paintings and drawings on walls and ceilings of a cave in Lascaux, France. See Figure 3.1.10. Use the information on page 86 to determine the approximate age of a piece of burned wood, if it was found that 8

Differential Equations with Boundary-Value Problems, 8
The Shroud of Turin, which shows the negative image of the body of a man who appears to have been crucified, is believed by many to be the burial shroud of Jesus of Nazareth. See Figure 3.1.11. In 1988 the Vatican granted permission to have the shroud carbon-dated. Three independent scientific labora

Differential Equations with Boundary-Value Problems, 8
A thermometer is removed from a room where the temperature is 70 F and is taken outside, where the air temperature is 10 F. After one-half minute the thermometer reads 50 F. What is the reading of the thermometer at t 1 min? How long will it take for the thermometer to reach 15 F?

Differential Equations with Boundary-Value Problems, 8
A thermometer is taken from an inside room to the outside, where the air temperature is 5 F. After 1 minute the thermometer reads 55 F, and after 5 minutes it reads 30 F. What is the initial temperature of the inside room?

Differential Equations with Boundary-Value Problems, 8
A small metal bar, whose initial temperature was 20 C, is dropped into a large container of boiling water. How long will it take the bar to reach 90 C if it is known that its temperature increases 2 in 1 second? How long will it take the bar to reach 98 C?

Differential Equations with Boundary-Value Problems, 8
Two large containers A and B of the same size are fille with different fluids. The fluids in containers A and B are maintained at 0 C and 100 C, respectively. A small metal bar, whose initial temperature is 100 C, is lowered into container A. After 1 minute the temperature of the bar is 90 C. After 2

Differential Equations with Boundary-Value Problems, 8
A thermometer reading 70 F is placed in an oven preheated to a constant temperature. Through a glass window in the oven door, an observer records that the thermometer reads 110 F after minute and 145 F after 1 minute. How hot is the oven?

Differential Equations with Boundary-Value Problems, 8
At t 0 a sealed test tube containing a chemical is immersed in a liquid bath. The initial temperature of the chemical in the test tube is 80 F. The liquid bath has a controlled temperature (measured in degrees Fahrenheit) given by Tm(t) 100  40e0.1t , t 0, where t is measured in minutes. (a) Assume

Differential Equations with Boundary-Value Problems, 8
A dead body was found within a closed room of a house where the temperature was a constant 70 F. At the time of discovery the core temperature of the body was determined to be 85 F. One hour later a second mea- 1 2 surement showed that the core temperature of the body was 80 F. Assume that the time o

Differential Equations with Boundary-Value Problems, 8
The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modifi cation of (2) is where k  0 and Tm is a constant. Suppose that two cups A and B are filled with coffee at the same time. Initially, the temperature of the coffee is 150 F. The exposed surface

Differential Equations with Boundary-Value Problems, 8
A tank contains 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 4 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t

Differential Equations with Boundary-Value Problems, 8
Solve Problem 21 assuming that pure water is pumped into the tank.

Differential Equations with Boundary-Value Problems, 8
A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. The wellmixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t

Differential Equations with Boundary-Value Problems, 8
In Problem 23, what is the concentration c(t) of the salt in the tank at time t? At t 5 min? What is the concentration of the salt in the tank after a long time, that is, as t : ? At what time is the concentration of the salt in the tank equal to one-half this limiting value?

Differential Equations with Boundary-Value Problems, 8
Solve Problem 23 under the assumption that the solution is pumped out at a faster rate of 10 gal/min. When is the tank empty?

Differential Equations with Boundary-Value Problems, 8
Determine the amount of salt in the tank at time t in Example 5 if the concentration of salt in the inflow is variable and given by cin(t) 2  sin(t4) lb/gal. Without actually graphing, conjecture what the solution curve of the IVP should look like. Then use a graphing utility to plot the graph of th

Differential Equations with Boundary-Value Problems, 8
A large tank is partially filled with 100 gallons of flui in which 10 pounds of salt is dissolved. Brine containing pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at a slower rate of 4 gal/min. Find the number of pounds of salt in t

Differential Equations with Boundary-Value Problems, 8
In Example 5 the size of the tank containing the salt mixture was not given. Suppose, as in the discussion following Example 5, that the rate at which brine is pumped into the tank is 3 gal/min but that the wellstirred solution is pumped out at a rate of 2 gal/min. It stands to reason that since brin

Differential Equations with Boundary-Value Problems, 8
A 30-volt electromotive force is applied to an LR-series circuit in which the inductance is 0.1 henry and the resistance is 50 ohms. Find the current i(t) if i(0) 0. Determine the current as t : .

Differential Equations with Boundary-Value Problems, 8
Solve equation (7) under the assumption that E(t) E0 sin vt and i(0) i0.

Differential Equations with Boundary-Value Problems, 8
A 100-volt electromotive force is applied to an RCseries circuit in which the resistance is 200 ohms and the capacitance is 104 farad. Find the charge q(t) on the capacitor if q(0) 0. Find the current i(t).

Differential Equations with Boundary-Value Problems, 8
A 200-volt electromotive force is applied to an RC-series circuit in which the resistance is 1000 ohms and the capacitance is 5 106 farad. Find the charge q(t) on the capacitor if i(0) 0.4. Determine the charge and current at t 0.005 s. Determine the charge as t : . 3

Differential Equations with Boundary-Value Problems, 8
An electromotive force is applied to an LR-series circuit in which the inductance is 20 henries and the resistance is 2 ohms. Find the current i(t) if i(0) 0.

Differential Equations with Boundary-Value Problems, 8
Suppose an RC-series circuit has a variable resistor. If the resistance at time t is given by R k1  k2t, where k1 and k2 are known positive constants, then (9) becomes

Differential Equations with Boundary-Value Problems, 8
Air Resistance In (14) of Section 1.3 we saw that a differential equation describing the velocity v of a falling mass subject to air resistance proportional to the instantaneous velocity is , where k 0 is a constant of proportionality. The positive direction is downward. (a) Solve the equation subjec

Differential Equations with Boundary-Value Problems, 8
How High?No Air Resistance Suppose a small cannonball weighing 16 pounds is shot vertically upward, as shown in Figure 3.1.12, with an initial velocity v0 300 ft/s. The answer to the question How high does the cannonball go? depends on whether we take air resistance into account. (a) Suppose air resi

Differential Equations with Boundary-Value Problems, 8
How High?Linear Air Resistance Repeat Problem 36, but this time assume that air resistance is proportional to instantaneous velocity. It stands to reason that the maximum height attained by the cannonball must be less than that in part (b) of Problem 36. Show this by supposing that the constant of pr

Differential Equations with Boundary-Value Problems, 8
Skydiving A skydiver weighs 125 pounds, and her parachute and equipment combined weigh another 35 pounds. After exiting from a plane at an altitude of 15,000 feet, she waits 15 seconds and opens her parachute. Assume that the constant of proportionality in the model in Problem 35 has the value k 0.5

Differential Equations with Boundary-Value Problems, 8
Evaporating Raindrop As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further assumptions that the rate at which the raindrop evaporates is proportional to its surface area and that air resistance is negligible, then a model for the velocity v(t) of the raindrop

Differential Equations with Boundary-Value Problems, 8
Fluctuating Population The differential equation dPdt (k cos t)P, where k is a positive constant, is a mathematical model for a population P(t) that undergoes yearly seasonal fluctuations. Solve the equation subject to P(0) P0. Use a graphing utility to graph the solution for different choices of P0.

Differential Equations with Boundary-Value Problems, 8
Population Model In one model of the changing population P(t) of a community, it is assumed that , where dBdt and dDdt are the birth and death rates, respectively. (a) Solve for P(t) if dBdt k1P and dDdt k2P. (b) Analyze the cases k1 k2, k1 k2, and k1  k2. 42. Co

Differential Equations with Boundary-Value Problems, 8
Constant-Harvest Model A model that describes the population of a fishery in which harvesting takes place at a constant rate is given by where k and h are positive constants. (a) Solve the DE subject to P(0) P0. (b) Describe the behavior of the population P(t) for increasing time in the three cases P

Differential Equations with Boundary-Value Problems, 8
Drug Dissemination A mathematical model for the rate at which a drug disseminates into the bloodstream is given by where r and k are positive constants. The function x(t) describes the concentration of the drug in the bloodstream at time t. (a) Since the DE is autonomous, use the phase portrait conce

Differential Equations with Boundary-Value Problems, 8
Memorization When forgetfulness is taken into account, the rate of memorization of a subject is given by , where k1 0, k2 0, A(t) is the amount memorized in time t, M is the total amount to be memorized, and M  A is the amount remaining to be memorized. (a) Since the DE is autonomous, use the phase

Differential Equations with Boundary-Value Problems, 8
Heart Pacemaker A heart pacemaker, shown in Figure 3.1.14, consists of a switch, a battery, a capacitor, and the heart as a resistor. When the switch S is at P, the capacitor charges; when S is at Q, the capacitor discharges, sending an electrical stimulus to the heart. In Problem 53 in Exercises 2.3

Differential Equations with Boundary-Value Problems, 8
Sliding Box (a) A box of mass m slides down an inclined plane that makes an angle u with the horizontal as shown in Figure 3.1.15. Find a differential equation for the velocity v(t) of the box at time t in each of the following three cases: (i) No sliding friction and no air resistance (ii) With slid

Differential Equations with Boundary-Value Problems, 8
Sliding BoxContinued (a) In Problem 46 let s(t) be the distance measured down the inclined plane from the highest point. Use dsdt v(t) and the solution for each of the three cases in part (b) of Problem 46 to find the time that it takes the box to slide completely down the inclined plane. A rootfin

Differential Equations with Boundary-Value Problems, 8
What Goes Up . . . (a) It is well known that the model in which air resistance is ignored, part (a) of Problem 36, predicts that the time ta it takes the cannonball to attain its maximum height is the same as the time td it takes the cannonball to fall from the maximum height to the ground. Moreover,