Questions and Answers of A First Course In Mathematical Modeling

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
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
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
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′ + 2y = 1– x–1In 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
``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
``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
``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
``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.
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
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
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
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
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
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
``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
``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
``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
``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
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,
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
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
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
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
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
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
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
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
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.

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