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mathematics
first course differential equations
A First Course in Differential Equations with Modeling Applications 11th edition Dennis G. Zill - Solutions
A tank in the form of a right circular cylinder standing on end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height h of water in the tank is described bydh/dt = - Ah/Aw √2gh,where Aw
Solve Problem 9 if 100 grams of chemical A is present initially. At what time is chemical C half formed?Data from problem 9Two chemicals A and B are combined to form a chemical C. The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of A and B not
Two chemicals A and B are combined to form a chemical C. The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially, there are 40 grams of A and 50 grams of B, and for each gram of B, 2 grams of A is used. It
(a) Suppose a = b = 1 in the Gompertz differential equation(7). Since the DE is autonomous, use the phase portrait concept of Section 2.1 to sketch representative solution curves corresponding to the cases P0 > e and 0 < P0 < e.(b) Suppose a = 1, b = -1 in (7). Use a new phase portrait to
Repeat Problem 6 in the case a = 5, b = 1, h = 7.Data from problem 6Investigate the harvesting model in Problem 5 both qualitatively and analytically in the case a = 5, b = 1, h = 25/4 . Determine whether the population becomes extinct infinite time. If so, find that time.
Investigate the harvesting model in Problem 5 both qualitatively and analytically in the case a = 5, b = 1, h = 25/4 . Determine whether the population becomes extinct infinite time. If so, find that time.Data from 5If a constant number h of fish are harvested from a fishery per unit
(a) If a constant number h of fish are harvested from a sherry per unit time, then a model for the population P(t) of the fishery at time t is given bydP/dt = P(a - bP) - h, P(0) = P0,where a, b, h, and P0 are positive constants. Suppose a = 5, b = 1, and h = 4. Since the DE is autonomous,
(a) Census data for the United States between 1790 and 1950 are given in the following table. Construct a logistic population model using the data from 1790, 1850, and 1910.(b) Construct a table comparing actual census population withthe population predicted by the model in part (a). Compute the
A model for the population P(t) in a suburb of a large city is given by the initial-value problemdP/dt = P(10-1 – 10-7 P), P(0) = 5000,where t is measured in months. What is the limiting value of the population? At what time will the population be equal to one half of this limiting value?
The number N(t) of people in a community who are exposed to a particular advertisement is governed by the logistic equation. Initially, N(0) = 500, and it is observed that N(1) = 1000. Solve for N(t) if it is predicted that the limiting number of people in the community who will see the
The number N(t) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problemdN/dt = N(1 - 0.0005N), N(0) = 1.(a) Use the phase portrait concept of Section 2.1 to predict how many supermarkets are expected to adopt the new procedure
(a) It is well known that the model in which air resistance is ignored, part (a)of Problem 36, predicts that the time tait takes the cannonball to attain its maximum height is the same as the time tdit takes the cannonball to fall from the maximum height to the ground. Moreover, the magnitude of
(a) In Problem 48 let s(t) be the distance measured down the inclined plane from the highest point. Use ds/dt = v(t) and the solution for each of the three cases in part (b) of Problem 48 to find the time that it takes the box to slide completely down the inclined plane. A root-finding
(a) A box of mass m slides down an inclined plane that makes an angle u with the horizontal as shown in Figure in the following. 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
A heart pacemaker, shown in the following figure, 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. We saw that during this time the
When forgetfulness is taken into account, the rate of memorization of a subject is given bydA/dt = k1(M - A) - k2A,where k1 > 0, k2 > 0, A(t) is the amount memorized intime t, M is the total amount to be memorized, and M - A is the amount remaining to be memorized.(a) Since the DE is
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 isdv/dt +
A model that describes the population of a fishery in which harvesting takes place at a constant rate is given bydP/dt = kP - h,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 P0
In one model of the changing population P(t) of a community, it is assumed thatdP/dt = dB/dt – dD/dt,where dB/dt and dDydt are the birth and death rates, respectively.(a) Solve for P(t) if dB/dt = k1P and dD/dt = k2P.(b) Analyze the cases k1 > k2, k1 = k2, and k1 < k2.
The differential equation dP/dt = (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.
Suppose of the rocket’s initial mass m0 that 50 kg is the mass of the fuel.(a) What is the burnout time tb, or the time at which all the fuel is consumed?(b) What is the velocity of the rocket at burnout?(c) What is the height of the rocket at burnout?(d) Why would you expect the rocket to attain
Suppose a small single-stage rocket of total mass m(t) is launched vertically, the positive direction is upward, the air resistance is linear, and the rocket consumes its fuel at a constant rate. In Problem 22 of Exercises 1.3 you were asked to use Newton’s second law of motion in the form given
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 during
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 proportionality is k =
Suppose a small cannonball weighing 16 pounds is shot vertically upward, as shown in figure, 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 resistance is ignored. If the
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 ism dv/dt = mg - kv,where k > 0 is a constant of proportionality. The positive direction is downward.(a)
An LR-series circuit has a variable inductor with the inductance defined byFind the current i(t) if the resistance is 0.2 ohm, the impressed voltage is E(t) = 4, and i(0) = 0. Graph i(t). |1 - 1, 0 10. [0,
An electromotive forceis 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. 120, 0 20
A 200-volt electromotive force is applied to an RC-series circuit in which the resistance is 1000 ohms and the capacitance is 5 × 10-6 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 → ∞.
A 100-volt electromotive force is applied to an RC-series circuit in which the resistance is 200 ohms and the capacitance is 10-4 farad. Find the charge q(t) on the capacitor if q(0) = 0. Find the current i(t).
Solve equation (7) under the assumption that E(t) = E0 sin ωt and i(0) = i0.
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 → ∞.
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 well-stirred solution is pumped out at a rate of 2 gal/min. It stands to reason that since
A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved. Brine containing 1/2 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
Determine the amount of salt in the tank at time t in Example 5 if the concentration of salt in the inflowis variable and given by cin(t) = 2 + sin(t/4) 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
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?Data from problem 23A 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
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?
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 well-mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t.
Solve Problem 21 assuming that pure water is pumped into the tank.Data from problem 21A 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
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.
The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modication of (2) isdT/dt = kS(T - Tm),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
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 measurement showed that the core temperature of the body was 80° F. Assume that the time
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 - 40e-0.1t, t ≥ 0, where t is measured in
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 12 minute and 145° F after 1 minute. How hot is the oven?
Two large containers A and B of the same size are filled 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
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?
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?
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?
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 the following figure. In 1988 the Vatican granted permission to have the shroud carbon-dated. Three independent
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 the following figure. Use the information on page 87 to determine the approximate age of a piece of burned wood, if it was
When interest is compounded continuously, the amount of money increases at a rate proportional to the amount S present at time t, that is, dS/dt = 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
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
(a) Consider the initial-value problem dA/dt = kA, A(0) = A0 as the model for the decay of a radioactive substance. Showthat, in general, the half-life T of the substance is T = 2(ln 2)/k.(b) Show that the solution of the initial-value problem in part (a) can be written A(t) = A02-t/T.(c) If a
Determine the half-life of the radioactive substance described in Problem 6.Data from problem 6Initially 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
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.
The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t and has a half-life 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?
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?
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?
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?
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?
Use the graphs in Problem 2 to approximate the times when the amounts x(t) and y(t) are the same, the times when the amounts x(t) and z(t) are the same, and the times when the amounts y(t) and z(t) are the same. Why does the time that is determined when the amounts y(t) and z(t) are the same make
After a mass m is attached to a spring, it stretches it s units and then hangs at rest in the equilibrium position as shown in the following figure (b). After the spring/mass system has been set in motion, let x(t) denote the directed distance of the mass beyond the equilibrium position. As
A cylindrical barrel s feet in diameter of weight w lb is floating in water as shown in the following figure (a). After an initial depression the barrel exhibits an up and down bobbing motion along a vertical line. Using following figure (b), determine a differential equation for the vertical
For high-speed motion through the air €”such as the skydiver shown in the following figure, falling before the parachute is opened €”air resistance is closer to a power of the instantaneous velocity v(t). Determine a differential equation for the velocity v(t) of a falling body
A series circuit contains a resistor and a capacitor as shown in the following figure. Determine a differential equation for the charge q(t) on the capacitor if the resistance is R, the capacitance is C, and the impressed voltage is E(t). R E
A series circuit contains a resistor and an inductor as shown in the following figure. Determine a differential equation for the current i(t) if the resistance is R, the inductance is L, and the impressed voltage is E(t). R.
The right-circular conical tank shown in the following figure loses water out of a circular hole at its bottom. Determine a differential equation for the height of the water h at time t > 0. The radius of the hole is 2 in., g = 32 ft/s2, and the friction/contraction factor introduced c =
Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cAhˆšgh, where c (0 < c < 1) is an empirical
What is the differential equation in Problem 10, if the well-stirred solution is pumped out at a faster rate of 3.5 gal/min?
Generalize the model given in equation (8) on page 24 by assuming that the large tank initially contains N0 number of gallons of brine, rin and rout are the input and output rates of the brine, respectively (measured in gallons per minute), cin is the concentration of the salt in the inflo , c(t)
Suppose that a large mixing tank initially holds 300 gallons of water is which 50 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min. If the
Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt have been dissolved. Pure water is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at the same rate. Determine a differential equation for the
A small single-stage rocket is launched vertically as shown in the following figure. Once launched, the rocket consumes its fuel, and so its total mass m(t) varies with time t > 0. If it is assumed that the positive direction is upward, air resistance is proportional to the instantaneous
In Problem 21, the mass m(t) is the sum of three different masses: m(t) = mp+ mv+ m(t), where mp is the constant mass of the payload, mvis the constant mass of the vehicle, and mf (t) is the variable amount of fuel.(a) Show that the rate at which the total mass m(t) of the rocket changes is the
By Newton’s universal law of gravitation the free-fall acceleration a of a body, such as the satellite shown in th4e following figure, falling a great distance to the surface is not the constant g. Rather, the acceleration a is inversely proportional to the square of the distance from the
Suppose a hole is drilled through the center of the Earth and a bowling ball of mass m is dropped into the hole, as shown in the following figure. Construct a mathematical model that describes the motion of the ball. At time t let r denote the distance from the center of the Earth to the mass m, M
In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t > 0. Determine a differential equation for the
In Problem 25 assume that the rate at which material is forgotten is proportional to the amount memorized in time t > 0. Determine a differential equation for the amount A(t) when forgetfulness is taken into account.
A drug is infused into a patient’s bloodstream at a constant rate of r grams per second. Simultaneously, the drug is removed at a rate proportional to the amount x(t) of the drug present at time t. Determine a differential equation for the amount x(t).
A person P, starting at the origin, moves in the direction of the positive x-axis, pulling a weight along the curve C, called a tractrix, as shown in the following figure. The weight, initially located on the y-axis at (0, s), is pulled by a rope of constant length s, which is kept taut throughout
Assume that when the plane curve C shown in the following figure is revolved about the x-axis, it generates a surface of revolution with the property that all light rays L parallel to the x-axis striking the surface are reflected to a single point O (the origin). Use the fact that the angle of
Reread the sentence following equation (3) and assume that Tm is a positive constant. Discuss why we would expect k < 0 in (3) in both cases of cooling and warming. You might start by interpreting, say, T(t) > Tm in a graphical manner.
Reread the discussion leading up to equation (8). If we assume that initially the tank holds, say, 50 lb of salt, it stands to reason that because salt is being added to the tank continuously for t > 0, A(t) should be an increasing function. Discuss how you might determine from the DE, without
The differential equation dp/dt = (k cost)P, where k is a positive constant, is a model of human population P(t) of a certain community. Discuss an interpretation for the solution of this equation. In other words, what kind of population do you think the differential equation describes?
As shown in following figure (a), a rightcircular cylinder partially filled with fluid is rotated with a constant angular velocity ω about a vertical y-axis through its center. The rotating fluid forms a surface of revolution S. To identify S, we first establish a coordinate system
Falling Body In Problem 23, suppose r = R + s, where s is the distance from the surface of the Earth to the falling body. What does the differential equation obtained in Problem 23 become when s is very small in comparison to R? [Hint: Think binomial series for(R + s)-2 = R-2 (1 + s/R)-2.]Data from
In meteorology the term virga refers to falling raindrops or ice particles that evaporate before they reach the ground. Assume that a typical raindrop is spherical. Starting at some time, which we can designate as t = 0, the raindrop of radius r0 falls from rest from a cloud and begins to
The “snowplow problem” is a classic and appears in many differential equations texts, but it was probably made famous by Ralph Palmer Agnew:One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour . What time
Suppose that dP/dt = 0.15P(t) represents a mathematical model for the growth of a certain cell culture, where P(t) is the size of the culture (measured in millions of cells) at time t > 0 (measured in hours). How fast is the culture growing at the time when the size of the culture reaches 2
Fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol c1 and has the form dy/dx = f (x, y). The symbol c1 represents a constant.1. d/dx c1e10x2. d/dx (5 + c1e-2x)
Fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1and c2and has the form F(y, y'' ) 0. The symbols c1, c2, and k represent constants.1. 2. d? drz (C1 cos kx + c2 sin kx) dx2 d? 5(c, cosh kx + c2 sinh kx) dx2
Compute y' and y'' and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y' y'') = 0. The symbols c1 and c2 represent constants.1. y = c1ex + c2xex2. y = c1ex cos x + c2ex sin x
Match each of the given differential equations with one or more of these solutions:(a) y = 0, (b) y = 2, (c) y = 2x, (d) y = 2x2.1. xy' = 2y2. y' = 23. y' = 2y 44. xy' = y5. y'' + 9y = 186. xy'' y' = 0
Determine by inspection at least one solution of the given differential equation.1. y'' = y'2. y' = y(y 3)
Interpret each statement as a differential equation.1. On the graph of y = ϕ(x) the slope of the tangent line at a point P(x, y) is the square of the distance from P(x, y) to the origin.2. On the graph of y = ϕ(x) the rate at which the slope changes with respect to x at a point P(x, y) is the
(a) Give the domain of the function y = x2/3.(b) Give the largest interval I of definition over which y = x2/3 is solution of the differential equation 3xy' 2y = 0.
(a) Verify that the one-parameter family y2 2y = x2 x + c is an implicit solution of the differential equation (2y 2)y' = 2x 1.(b) Find a member of the one-parameter family in part (a) that satisfies the initial condition y(0) = 1.(c) Use your result in part (b) to find an explicit function
Given that y = x 2/x is a solution of the DE xy' + y = 2x. Find x0 and the largest interval I for which y(x) is a solution of the first-order IVP xy' + y = 2x, y(x0) = 1.
Suppose that y(x) denotes a solution of the first-order IVP y' = x2 + y2, y(1) = 1 and that y(x) possesses at least a second derivative at x = 1. In some neighborhood of x = 1 use the DE to determine whether y(x) is increasing or decreasing and whether the graph y(x) is concave up or concave down.
A differential equation may possess more than one family of solutions.(a) Plot different members of the families y = ϕ1(x) = x2 + c1 and y = ϕ2(x) = -x2 + c2.(b) Verify that y = ϕ1(x) and y = φ2(x) are two solutions of the nonlinear first-order differential equation (y')2 = 4x2.(c) Construct a
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