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a first course in mathematical modeling

A First Course In Mathematical Modeling 5th Edition Frank R. Giordano, William P. Fox, Steven B. Horton - Solutions

Solve y′ = x(1–y2), where –1 < y < 1.
Consider the following model for the cooling of a hot cup of soup:where Tm (0) = α. Here Tm is the temperature of the soup at any time t > 0,βis the constant temperature of the surrounding medium, is the initial temperature of the soup, and k is a constant of proportionality depending on the
In Section 11.1 we developed the following model for population growth in a limited environment:where P(t0) = P0. Here P denotes the population at any time t > 0, M is the carrying capacity of the environment, and r is a proportionality constant. Let us solve this model.
dy/dx = y2– 2y + 1In Problems 1–8, solve the separable differential equation using u-substitution.
dy/dx = √y cos2 √yIn Problems 1–8, solve the separable differential equation using u-substitution.
y′ = 3y (x + 1)2/y–1In Problems 1–8, solve the separable differential equation using u-substitution.
y y′ = sec y2 sec2 xIn Problems 1–8, solve the separable differential equation using u-substitution.
y′ = xe y √x–2In Problems 1–8, solve the separable differential equation using u-substitution.
sec x dy + x cos2 y dx = 0In Problems 9–16, solve the separable differential equation using integration by parts.
2x2 dx–3√y csc x dy = 0In Problems 9–16, solve the separable differential equation using integration by parts.
y′= e y/xyIn Problems 9–16, solve the separable differential equation using integration by parts.
y′ = xex–y csc yIn Problems 9–16, solve the separable differential equation using integration by parts.
y′ = e–y ln (1/x)In Problems 9–16, solve the separable differential equation using integration by parts.
y′ = y2 tan–1 xIn Problems 9–16, solve the separable differential equation using integration by parts.
y′ = y sin–1 xIn Problems 9–16, solve the separable differential equation using integration by parts.
sec (2x + 1) dy + 2xy–1 dx = 0In Problems 9–16, solve the separable differential equation using integration by parts.
y′ = (y2 –1) x–1In Problems 17–24, solve the separable differential equation using partial fractions.
x(x–1) dy–y dx = 0In Problems 17–24, solve the separable differential equation using partial fractions.
y′= (y +1)2/x2 + x– 2In Problems 17–24, solve the separable differential equation using partial fractions.
Find the general solution of:
Find the general solution of:
Find the solution of:
We now return to the problem of water pollution of a large lake introduced at the beginning of this section. Suppose a large lake that was formed by damming a river holds initially 100 million gallons of water. Because a nearby agricultural field was sprayed with a pesticide, the water has become
y′– 3y = exIn Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
2y′ – y = xex/2In Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
xy′ – y = 2x ln xIn Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
y′ = y –e2xIn Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
x2 dy/dx+ xy = 2In Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
(1 + x)dy/dx+ y =√xIn Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
x2 dy + xy dx = (x –1)2 dxIn Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
(1 + ex) dy +(ye x + e–x) dx = 0In Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
e–y dx + (e–y x –4y) dy = 0In Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
(x + 3y2) dy + y dx = 0In Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
y dx + (3x– y–2 cos y) dy = 0, y > 0In Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
y′ + 4y = 1, y (0) = 1In Problems 16–20, solve the initial value problem.
dy/dx+ 3x2y = x2, y (0) = –1In Problems 16–20, solve the initial value problem.
x dy + (y –cos x) dx = 0, y (π /2) = 0In Problems 16–20, solve the initial value problem.
xy′ + (x – 2) y = 3x3e–x, y (1) = 0In Problems 16–20, solve the initial value problem.
Find a dimensionless product relating the torque τ (ML2T–2) produced by an automobile engine, the engine's rotation rate Ψ(T –1), the volume V of air displaced by the engine, and the air density ρ .
The various constants of physics often have physical dimensions (dimensional constants) because their values depend on the system in which they are expressed. For example, Newton's law of gravitation asserts that the attractive force between two bodies is proportional to the product of their masses
Certain stars, whose light and radial velocities undergo periodic vibrations, are thought to be pulsating. It is hypothesized that the period t of pulsation depends on the star's radius r, its mass m, and the gravitational constant G. (See Problem 3 for the dimension of G.) Express t as a product
In checking the dimensions of an equation, you should note that derivatives also possess dimensions. For example, the dimension of ds/dt is LT –1 and the dimension of d2s/dt2 is LT–2, where s denotes distance and t denotes time. Determine whether the equation:for the time rate of change of
For a body moving along a straight-line path, if the mass of the body is changing over time, then an equation governing its motion is given by:where m is the mass of the body, v is the velocity of the body, F is the total force acting on the body, dm is the mass joining or leaving the body in the
Assume the force F opposing the fall of a raindrop through air is a product of viscosity μ , velocity v, and the diameter r of the drop. Assume that density is neglected. Find:
Predict the time of revolution for two bodies of mass m1 and m2 in empty space revolving about each other under their mutual gravitational attraction.
A projectile is red with initial velocity v at an angle θwith the horizon. Predict the range R.
Consider an object that is falling under the influence of gravity. Assume that air resistance is negligible. Using dimensional analysis, find the speed v of the object after it has fallen a distance s. Let v = f (m, g, s), where m is the mass of the object and g is the acceleration due to gravity.
One would like to know the nature of the drag forces experienced by a sphere as it passes through a fluid. It is assumed that the sphere has a low speed. Therefore, the drag force is highly dependent on the viscosity of the fluid. The fluid density is to be neglected. Use the dimensional analysis
The volume flow rate q for laminar flow in a pipe depends on the pipe radius r, the viscosity μ of the fluid, and the pressure drop per unit length dp/dz. Develop a model for the flow rate q as a function of r,μ , and dp/dz.
The power P delivered to a pump depends on the specific weight w of the fluid pumped, the height h to which the fluid is pumped, and the fluid flow rate q in cubic feet per second. Use dimensional analysis to determine an equation for power.
Find the volume flow rate dV/dt of blood flowing in an artery as a function of the pressure drop per unit length of artery, the radius r of the artery, the blood density ρ, and the blood viscosity μ.
The speed of sound in a gas depends on the pressure and the density. Use dimensional analysis to find the speed of sound in terms of pressure and density.
Under appropriate conditions, all three models for the pendulum imply that t is proportional to √r. Explain how the conditions distinguish among the three models by considering how m must vary in each case.
a. Show that the products П1,Π2,Π3,Π4 for the refined explosion model presented in the text are dimensionless products.b. Assume ρ is essentially constant for the soil being used and restrict the explosive to a specific type, say TNT. Under these conditions, ρ/δ=is essentially constant,
Consider a zero-depth burst, spherical explosive in a soil medium. Assume the value of the crater volume V depends on the explosive size, energy yield, and explosive energy, as well as on the strength Y of the soil (considered a resistance to pressure with dimensions ML–1T–2), soil density ,
Repeat Problem 4 for:Data from problem 4For the explosion process and material characteristics discussed in Problem 3, consider:
An experiment consists of dropping spheres into a tank of heavy oil and measuring the times of descent. It is desired that a relationship for the time of descent be determined and verified by experimentation. Assume the time of descent is a function of mass m, gravity g, radius r, viscosity μ, and
A windmill is rotated by air flow to produce power to pump water. It is desired to find the power output P of the windmill. Assume that P is a function of the density of the air ρ, viscosity of the air μ , diameter of the windmill d, wind speed v, and the rotational speed of the windmill ω
For a sphere traveling through a liquid, assume that the drag force FD is a function of the fluid density ρ , fluid viscosity μ , radius of the sphere r, and speed of the sphere v. Use dimensional analysis to find a relationship for the drag force:
Two smooth balls of equal weight but different diameters are dropped from an airplane. The ratio of their diameters is 5. Neglecting compressibility (assume constant Mach number), what is the ratio of the terminal velocities of the balls? Are the flows completely similar?
Analyze the effect on the arms race of each of the following strategies:a. Country X increases the accuracy of its missiles by using a better guidance system.b. Country X increases the payload (destructive power) of its missiles without sacrificing accuracy.c. Country X is able to retarget its
Discuss the appropriateness of the assumptions used in developing the nuclear arms race model. What is the effect on the number of missiles if each country believes the other country is also following the friendly strategy? Is disarmament possible?
Develop a graphical model based on the assumption that each side is following the enemy strategy. That is, each side desires a first-strike capability for destroying the missile force of the opposing side. What is the effect on the arms race if Country X now introduces antiballistic missiles?
Discuss how you might go about validating the nuclear arms race model. What data would you collect? Is it possible to obtain the data?
Build a numerical solution to Equations (15.8).a. Graph your results.b. Is an equilibrium value reached?c. Try other starting values. Do you think the equilibrium value is stable?d. Explore other values for the survival coefficients of Countries X and Y. Describe your results.Equation 15.8
Recall from Section 15.1 that an equilibrium value for the arms race requires that xn+1= xn and yn+1 = yn simultaneously. Is there an equilibrium value for Equations (15.7)? If so, find it.Equation 15.7
``The Distribution of Resources,'' by Harry M. Schey, UMAP 6062 (one module). The author investigates a graphical model that can be used to measure the distribution of resources. The module provides an excellent review of the geometric interpretation of the derivative as applied to the economics
``Nuclear Deterrence,'' by Harvey A. Smith, UMAP327. The author analyzes the stability of the arms race, assuming objectives similar to those suggested by General Taylor. The module develops analytic models using probabilistic arguments. An understanding of elementary probability is required.For
``The Geometry of the Arms Race,'' by Steven J. Brams, Morton. Davis, and Philip Staffin, Jr., UMAP 311. This module analyzes the possibilities of both parties disarming by introducing elementary game theory. Interesting conclusions are based on Country's. ability to detect Country Y 's
``The Richardson Arms Race Model,'' by Dina A. Zines, John V. Gillespie, and G. S. Tahim, UMAP 308. A model is constructed on the basis of the classical assumptions of Lewis Fry Richardson. Difference equations are introduced.For Projects 1-4, complete the requirements in the referenced UMAP
Justify mathematically, and interpret economically, the graphical model for the theory of the firm given in Figure 15.18. What are the major assumptions on which the model is based?Figure 15.8
Show that for total profit to reach a relative minimum, MR = MC and MC′ < MR′.
Suppose the large competitive industry is the oil industry, and the firm within that industry is a gasoline station. How well does the model depicted in Figure 15.17 reflect the reality of that situation? How would you adjust the graphical model to make improvements?Figure 15.17
Verify the result that the marginal revenue of the q+1st unit equals the price of that unit minus the loss in revenue on previous units resulting from price reduction.
Show that when the demand curve is very steep, a tax added to each item sold will fall primarily on consumers. Now show that when the demand curve is more nearly horizontal, the tax is paid mostly by the industry. What if the supply curve is very steep? What if the supply curve is nearly horizontal?
Consider the oil industry. Discuss the conditions for which the demand curve will be steep near the equilibrium. What are the situations for which the demand curve will be more horizontal (or flat)?
Criticize the following quotation:The effect of a tax on a commodity might seem at first sight to be an advance in price to the consumer. But an advance in price will diminish the demand, and a reduced demand will send the price down again. It is not certain, therefore, after all, that the tax will
Suppose the government pays producers a subsidy for each unit produced instead of levying a tax. Discuss the effect on the equilibrium point of the supply and demand curves. What happens to the new price and the new quantity? Discuss how the proportion of the benets to the consumer and to the
Consider the graphical model in Figure 15.27. Argue that if the demand curve fails to shift significantly to the left, an increase in the equilibrium quantity could occur after the crisis.Figure 15.27
Consider the situation in which demand is a fixed curve but there is an increase in supply, so the supply curve shifts downward. Discuss how the slope of the demand curve affects the change in price and the change in quantity: How does the price change, and when does it change the most? When does
Criticize the graphical model of the oil industry. Name some major factors that have been neglected. Which of the underlying assumptions are not satisfied by the crisis situation? Did the graphical model help you identify some of the key factors and their interactions? How could you adjust the
``Differentiation, Curve Sketching, and Cost Functions,'' by Christopher H. Nevison, UMAP 376. In this module costs and revenue for a firm are discussed using elementary calculus. The author discusses several of the economic ideas presented in this chapter.For Projects 1-4, complete the
``Price Discrimination and Consumer Surplus: An Application of Calculus to Economics,'' by Christopher H. Nevison, UMAP294. The topics in this module are analyzed in a competitive market, and two-tier price discrimination is also discussed. The module examines several of the economic ideas
``Economic Equilibrium: Simple Linear Models,'' by Philip M. Tuchinsky, UMAP 208.In this module linear supply and demand functions are constructed, and the equilibrium market position is analyzed for an industry producing one product. The result is then extended to n products. The author concludes
``I Will If You Will ... A Critical Mass Model,'' by Jo Anne S. Growney, UMAP 539. A graphical model is presented to treat the problems of individual behavior in a group when the individual makes a choice dependent on his or her perception of the behavior of fellow group members. The model can
Consider an intersection of two one-way streets controlled by a traffic light. Assume that between 5 and 15 cars (varying probabilistically) arrive at the intersection every 10 sec in direction 1, and that between 6 and 24 cars arrive every 10 sec going in direction 2.Suppose that 36 cars per 10
Modify Steps 20-32 in the elevator simulation algorithm so that loading of the first available elevator commences immediately upon its return. Thus, if TIME > return j so that elevator j is available for loading, then loading commences at time return j rather than TIME. Consider how you will now
Find a building in your local area that has from 4 to 12 floors that are serviced by 1-4 elevators. Collect data for the interarrival times (and, possibly, floor destinations) of the customers during a busy hour (e.g., the morning rush hour), and build the interarrival and destination submodels
Find the integral ∫ xe x dx by the tabular method.
Integrate ∫x2e2xdx by the tabular method.
Integrate ∫ex sin x dx.
Find the first three approximations y1, y2, y3 using Euler's method for the initial value problem:
Let's discuss again the savings certificate example investigated in Chapter 1 as a discrete dynamical system. Here, we consider the value of a certicate initially worth $1000 that accumulates annual interest at 12% compounded continuously (rather than 1% each month as in Example 1 of Section 1.1).
Construct and perform a Monte Carlo simulation of a darts game. The rules are:Make an assumption about the distribution of how the darts hit on the board. Write an algorithm, and code it in the computer language of your choice. Run 1000 simulations to determine the mean score for throwing five
Construct and perform a Monte Carlo simulation of blackjack (also called twenty-one). The rules of blackjack are as follows: Most casinos use six or eight decks of cards when playing this game to inhibit ``card counters.'' You will use two decks of cards in your simulation (104 cards
Write a program or use a spreadsheet to find the approximate area or volumes in Problems 3–7 in Section 5.1.Data from problem 3Using Monte Carlo simulation, write an algorithm to calculate an approximation to π by considering the number of random points selected inside the quarter circle:Data
Write a program to generate 1000 integers between 1 and 5 in a random fashion so that 1 occurs 22% of the time, 2 occurs 15% of the time, 3 occurs 31% of the time, 4 occurs 26% of the time, and 5 occurs 6% of the time. Over what interval would you generate the random numbers? How do you decide
Write a computer program to generate uniformly distributed random integers in the interval m Here, floor [p] means the largest integer not exceeding p. For most choices of Y, the numbersX1,X2.... form a sequence of (pseudo)random integers as desired. One possible recommended choice is Y=
Refer to ``Random Numbers'' by Mark D. Myerson, UMAP 590. This module discusses methods for generating random numbers and presents tests for determining the randomness of a string of numbers. Complete this module and prepare a short report on testing for randomness.
Complete the requirement for UMAP module 269, ``Monte Carlo: The Use of Random Digits to Simulate Experiments,'' by Dale T. Hoffman. The Monte Carlo technique is presented, explained, and used to find approximate solutions to several realistic problems.Simple experiments are included for student

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