Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Differential Equations

Consider the autonomous differential equation dy/dt=f(y) where the graph of f(y) is given below. (a) Sketch the phase line for this equation and identify the equilibrium points assinks, sources, or nodes. (b) Give a rough sketch of the slope field that corresponds to this equation. (c) Give rough sketches of the graphs of the solutions that satisfy the initial conditions y(0)=3,y(0)=0,y(0)=1, and y(0)=2 .43RE 44RE 45RE 46RE 47RE 48RE Eight differential equations and four slope fields are given below. Determine the equation that corresponds to each slope field and state briefly how you know your choice is correct. You should do this exercise without using technology. (i) dydt=t1 (ii) dydt=1y2 (iii) dydt=yt2 (iv) dydt=1t (v) dydt=1y (vi) dydt=y+t2 (vii) dydt=tyt (viii) dydt=y21 50RE 51RE Consider the differential equation dy/dt=2ty2 . (a) Calculate its general solution. (b) Find all values of y0 such that the solution to the initial-value problem dydt=2ty2,y(1)=y0 ,does not blow up (or down) in finite time. In other words, find all y0 such thatthe solution is defined for all real t.53RE A 1000-galIon tank initially contains a mixture of 450 gallons of cola and 50 gallons of cherry syrup. Cola is added at the rate of 8 gallons per minute, and cherry syrupis added at the rate of 2 gallons per minute. At the same time, a well mixed solution of cherry cola is withdrawn at the rate of 5 gallons per minute. What percentage ofthe mixture is cherry syrup when the tank is full?Exercises 1-6 refer to the following systems of equations: (i) dxdt=10x(1x 10)20xydydt=5y+xy20 (ii) dxdt=0.3xxy100dydt=15y(1y 50)+25xy 1. In one of these systems, the prey are very large animals and the predators are very small animals, such as elephants and mosquitoes. Thus it takes many predators to cat one prey, but each prey eaten is a tremendous benefit for the predator population. The other system has very large predators and very small prey. Determine which system is which and provide a justification for your answer.Exercises 1-6 refer to the following systems of equations: (i) dxdt=10x(1x 10)20xydydt=5y+xy20 (ii) dxdt=0.3xxy100dydt=15y(1y 50)+25xy 2. Find all equilibrium points for the two systems. Explain the significance of these points in terms of the predator and prey populations.Exercises 1-6 refer to the following systems of equations: (i) dxdt=10x(1x 10)20xydydt=5y+xy20 (ii) dxdt=0.3xxy100dydt=15y(1y 50)+25xy 3. Suppose that the predators arc extinct at time t0=0 . For each system, verify that the predators remain extinct for all time.Exercises 1-6 refer to the following systems of equations: (i) dxdt=10x(1x 10)20xydydt=5y+xy20 (ii) dxdt=0.3xxy100dydt=15y(1y 50)+25xy 4. For each system, describe the behavior of the prey population if the predators are extinct. (Sketch the phase line for the prey population assuming that the predators are extinct, and sketch the graphs of the prey population as a function of time for several solutions. Then interpret these graphs for the prey population.)5E 6E Consider the predator-prey system dRdt=2(1R3)RRFdFdt=2F+4RF The figure to the right shows a computer generated plot of a solution curve for this system in the RF-plane. (a) Describe the fate of the prey (R)and predator (F)populations based on this image. (b) Confirm your answer using HPGSystemSolver.Consider the predator-prey system dRdt=2R(1R 2.5)1.5RFdFdt=F+0.8RF and the solution curves in the phase plane on the right. (a) Sketch the R(t)-and F(t)-graphs for the solutions with initial points A, B, C, and D. (b) Interpret each solution curve in terms of the behavior of the populations over time. (c) Confirm your answer using HPGSystemSolver.Exercises 9-14 refer to the predator-prey and the modified predator-prey systems discussed in the text (repeated here for convenience): (i) dRdt=2R1.2RFdFdt=F+0.9RF (ii) dRdt=2R(1R2)1.2RFdFdt=F+0.9RF 9. How would you modify these systems to include the effect of hunting of the prey at a rate of units of prey per unit of time?Exercises 9-14 refer to the predator-prey and the modified predator-prey systems discussed in the text (repeated here for convenience): (i) dRdt=2R1.2RFdFdt=F+0.9RF (ii) dRdt=2R(1R2)1.2RFdFdt=F+0.9RF 10. How would you modify these systems to include the effect of hunting of the predators at a rate proportional to the number of predators?Exercises 9-14 refer to the predator-prey and the modified predator-prey systems discussed in the text (repeated here for convenience): (i) dRdt=2R1.2RFdFdt=F+0.9RF (ii) dRdt=2R(1R2)1.2RFdFdt=F+0.9RF 11. Suppose the predators discover a second, unlimited source of food, but they still prefer to cat prey when they can catch them. How would you modify these systems to include this assumption?12E 13E Exercises 9-14 refer to the predator-prey and the modified predator-prey systems discussed in the text (repeated here for convenience): (i) dRdt=2R1.2RFdFdt=F+0.9RF (ii) dRdt=2R(1R2)1.2RFdFdt=F+0.9RF 14. Suppose prey move out of an area at a rate proportional to the number of predator in the area. How would you modify these systems to take this into account?15E Consider the system of predator-prey equations dRdt=2(1R3)RRFdFdt=16F+4RF The figure below shows a computer-generated plot of a solution curve for this system in the RF-plane. (a) What can you say about the fate of the rabbit R and fox F populations based on this image? (b) Confirm your answer using HPGSystemSolver.Pesticides that kill all insect species are not only bad for the environment, but they can also be inefficient at controlling pest species. Suppose a pest insect species in a particular field has population R(t)a time t,and suppose its primary predator is another insect species with population F(t)at time t. Suppose the populations of these species are accurately modeled by the system dRdt=2R1.2RFdFdt=F+0.9RF studied in this section. Finally, suppose that at time t= 0 a pesticide is applied to the field that reduces both the pest and predator populations to very small hut nonzero numbers. (a) Using Figures 2.3 and 2.5,predict what will happen as t increases to the population of the pest species. (b) Write a short essay, in nontechnical language, warning of the possibility of the ‘paradoxical” effect that pesticide application can have on pest populations.Some predator species seldom capture healthy adult prey, eating only injured or weak prey. Because weak prey consume resources are not as successful at reproduction. the harsh reality is that their removal from the population increases prey population. Discuss how you would modify a predator-prey system to model this sort of interaction.19E Consider the initial-value problem d2ydt2+kmy=0 for the motion of a simple harmonic oscillator. (a)Consider the function y(t)=cost . Under what conditions on is y(t) a solution? (b) What initial condition (t=0) in the yv-p1ane corresponds to this solution. (c) In terms of k and m, what is the period of this solution? (d) Sketch the solution curve (in the yv-plane) associated to this solution. [Hint: Consider the quantity y2+(v/)2 .]A mass weighing 12 pounds stretches a spring 3 inches. What is the spring constant for this spring?A mass weighing 4 pounds stretches a spring 4 inches. (a) Formulatean initial-value problem that corresponds to the motion of this und amped mass-spring system lithe mass is extended 1 foot from its rest position and released (with no initial velocity). (b) Using the result of Exercise 20, find the solution of this initial-value problem.Do the springs in an “extra firm’ mattress have a large spring constant or a small spring constant?Consider a vertical mass-spring system as shown in the figure below. Before the mass is placed on the end of the spring, the spring has a natural length. After the mass is placed on the end of the spring, the system has a new equilibrium position, which corresponds to the position where the force on the mass due to gravity is equal to the force on the mass due to the spring. (a) Assuming that the only forces acting on the mass are the force due to gravity and the force of the spring, formulate two different (but related) second-order differential equations that describe the motion of the mass. For one equation, let the position y(t)be measured from the point at the end of the spring when it hangs without the mass attached. For the other equation, let y2(t) be measured from the equilibrium position once the mass is attached to the spring. (b) Rewrite these two second-order equations as first-order systems and calculate their equilibrium points. Interpret your results in terms of the mass-spring system. (c) Given a solution y1(t) to one system, how can you produce a solution y2(t)to the second system? (d) Which choice of coordinate system, y1or y2do you prefer? Why?Exercises 25—30 refer to a situation in which models similar to the predator-prey population models arise. Suppose A and B represent two substances that can combine to form a new substance C (chemists would write A + B C). Suppose we have a container with a solution containing low Concentrations of substances A and B, and A and B molecules react only when they happen to come close to each other. If a(t) and b(t) represent the amount of A and B in the solution, respectively, then the chance that a molecule of A is close to a molecule of B at time t is proportional to the product a(t).b(t). Hence the rate of reaction of A and B to form C is proportional to ab. Suppose C precipitates out of the solution as soon as it is formed, and the solution is always kept well mixed. 25. Write a system of differential equations that models the evolution of a(t) and b(t). Be sure to identify and describe any parameters you introduce.26E 27E 28E 29E Exercises 25—30 refer to a situation in which models similar to the predator-prey population models arise. Suppose A and B represent two substances that can combine to form a new substance C (chemists would write A + B C). Suppose we have a container with a solution containing low Concentrations of substances A and B, and A and B molecules react only when they happen to come close to each other. If a(t) and b(t) represent the amount of A and B in the solution, respectively, then the chance that a molecule of A is close to a molecule of B at time t is proportional to the product a(t).b(t). Hence the rate of reaction of A and B to form C is proportional to ab. Suppose C precipitates out of the solution as soon as it is formed, and the solution is always kept well mixed. 30. Suppose A and B are being added to the solution at constant (perhaps unequal) rates, and, in addition to the A + B C reaction, a reaction A + 2B D can occur when two B and one A molecules are close. Suppose substance D precipitates out of the solution. How would you modify your system of equations o include these assumptions?1E 2E 3E 4E 5E 6E 7E Convert the second-order differential equation 1 d2ydt2+2y=0 into a first-order system in terms of y and v where v = dy/dt. (a) Determine the vector field associated with the first-order system. (b) Sketch enough vectors in the vector field to get a sense of its geometric structure. (You should do this part of the exercise without the use of technology.) (c) Use HPGSystemSolver to sketch the associated direction field. (d) Make a rough sketch of the phase portrait of the system and confirm your answer using HPGSystemsolver. (e) Briefly describe the behavior of the solutions.9E Consider the system dxdt=2x+ydydt=2y and its corresponding direction field. (a) Sketch a number of different solution curves on the phase plane. (b) Describe the behavior of the solution that satisfies the initial condition (x0, y0) = (0, 2).Eight systems of differential equations and four direction fields are given below. Determine the system that corresponds to each direction field and state briefly how you know your choice is correct. You should do this exercise without using technology. (i) dxdt=xdydt=y1 (ii) dxdt=x21dydt=y (iii) dxdt=x+2ydydt=y (iv) dxdt=2xdydt=y (v) dxdt=xdydt=2y (vi) dxdt=x1dydt=y (vii) dxdt=x21dydt=y (viii) dxdt=x2ydydt=y Consider the modified predator-prey system dRdt=2R(1R2)1.2RFdFdt=F+0.9RF discussed in Section 2.1. Find all equilibrium solutions.In Exercises 13—18. (a) find the equilibrium points of the system. (b) using HPGSystemSolver, sketch the direction field and phase portrait of the system, and (c) briefly describe the behavior of typical solutions. 13. dxdt=4x7y+2dydt=3x+6y1 14E 15E In Exercises 13—18. (a) find the equilibrium points of the system. (b) using HPGSystemSolver, sketch the direction field and phase portrait of the system, and (c) briefly describe the behavior of typical solutions. 16. dxdt=ydydt=xx3y 17E In Exercises 13—18. (a) find the equilibrium points of the system. (b) using HPGSystemSolver, sketch the direction field and phase portrait of the system, and (c) briefly describe the behavior of typical solutions. 18. dxdt=4(x2y21)dydt=x(x2+y21)19E 20E Consider the four solution curves in the phase portrait and the four pairs of x(t)- and y(t)-graphs shown below. Match each solution curve with its corresponding pair of x(t)- and y(t)-graphs. Then on the x-axis mark the t-values that correspond o the distinguished points along the curve.22E 23E 24E 25E 26E 27E In Exercises 1—4, a harmonic oscillator equation for y(t) is given. (a) Using HPGSystemSolver, sketch the associated direction field. (b) Using the guess-and-test method described in this section, find two nonzero solutions that are not multiples of one another. (c) For each solution, plot both its solution curve in the yv-plane and its x(t)- and v(t)-graphs. 1. d2ydt2+7dydt+10y=0 In Exercises 1—4, a harmonic oscillator equation for y(t) is given. (a) Using HPGSystemSolver, sketch the associated direction field. (b) Using the guess-and-test method described in this section, find two nonzero solutions that are not multiples of one another. (c) For each solution, plot both its solution curve in the yv-plane and its x(t)- and v(t)-graphs. 2. d2ydt2+5dydt+6y=0 In Exercises 1—4, a harmonic oscillator equation for y(t) is given. (a) Using HPGSystemSolver, sketch the associated direction field. (b) Using the guess-and-test method described in this section, find two nonzero solutions that are not multiples of one another. (c) For each solution, plot both its solution curve in the yv-plane and its x(t)- and v(t)-graphs. 3. d2ydt2+4dydt+y=0 In Exercises 1—4, a harmonic oscillator equation for y(t) is given. (a) Using HPGSystemSolver, sketch the associated direction field. (b) Using the guess-and-test method described in this section, find two nonzero solutions that are not multiples of one another. (c) For each solution, plot both its solution curve in the yv-plane and its x(t)- and v(t)-graphs. 4. d2ydt2+6dydt+7y=0 5E In the damped harmonic oscillator, we assume that the coefficients m, b, and k are positive. However, the rationale underlying the guess-and-test method made no such assumption, and the same analytic technique can be used if some or all of the coefficients of the equation are negative. In Exercises 5 and 6, make the same graphs and perform the same calculations as were specified in Exercises 1—4. What is different in this case? 6. d2ydt2+dydt2y=0 Consider any damped harmonic oscillator equation md2ydt2+bdydt+ky=0 (a) Show that a constant multiple of any solution is another solution. (b) Illustrate this fact using the equation d2ydt2+3dydt+2y=0 discussed in the section. (c) How many solutions to the equation do you get if you use this observation along with the guess-and-test method described in this section?Consider any damped harmonic oscillator equation md2ydt2+bdydt+ky=0 (a) Show that the sum of any two solutions is another solution. (b) Using the result of part (a), solve the initial-value problem d2ydt2+3dydt+2y=0,y(0)=2,v(0)=3 (c) Using the result of part (a) in Exercise 7 along with the result of part (a) of this exercise, solve the initial-value problem d2ydt2+3dydt+2y=0,y(0)=3,v(0)=5 (d) How many solutions to the equation d2ydt2+3dydt+2y=0 do you get if you use the results of Exercise 7 and this exercise along with the guess-and-test method described in this section?In Exercises 9 and 10, we consider a mass sliding on a frictionless table between two walls that are 1 unit apart and connected to both walls with springs, as shown below. Let k1, and k2be the spring constants of the left and right spring, respectively, let m be the mass, and let b be the damping coefficient of the medium the spring is sliding through. Suppose L1and L2are the rest lengths of the left and right springs, respectively. 9. Write a second-order differential equation for the position of the mass at time t. [Hint: The first step is to pick an origin, that is. a point where the position is 0. The left-hand wall is a natural choice.]In Exercises 9 and 10, we consider a mass sliding on a frictionless table between two walls that are 1 unit apart and connected to both walls with springs, as shown below. Let k1, and k2be the spring constants of the left and right spring, respectively, let m be the mass, and let b be the damping coefficient of the medium the spring is sliding through. Suppose L1and L2are the rest lengths of the left and right springs, respectively. 10. (a) Convert the second-order equation of Exercise 9 into a first-order system. (b) Find the equilibrium point of this system. (c) Using your result from part (b), pick a new coordinate system and rewrite the system in terms of this new coordinate system. (d) How does this new system compare o the system for a damped harmonic oscillator?In Exercises 1-4, we consider the system dxdt=2x+2ydydt=x+3y For the given functions Y(t)=(x(t),y(t)) , determine if Y(t) is a solution system. 1. (x(t),y(t))=(2et,et)In Exercises 1-4, we consider the system dxdt=2x+2ydydt=x+3y For the given functions Y(t)=(x(t),y(t)) , determine if Y(t) is a solution system. 2.(x(t),y(t))=(3e2t+et,et+e4t)In Exercises 1-4, we consider the system dxdt=2x+2ydydt=x+3y For the given functions Y(t)=(x(t),y(t)) , determine if Y(t) is a solution system. 3. (x(t),y(t))=(2ete4t,et+e4t)In Exercises 1-4, we consider the system dxdt=2x+2ydydt=x+3y For the given functions Y(t)=(x(t),y(t)) , determine if Y(t) 4. (x(t),y(t))=(4et+e4t,2et+e4t)In Exercises 5-12, we consider the partially decoupled system dxdt=2x+2ydydt=y 5. Although we can use the method described in this section to derive the general solution to this system, why should we immediately know that Y(t)=(x(t),y(t))=(e2tet,e2t)is not a solution to the system?6E In Exercises 5-12, we consider the partially decoupled system dxdt=2x+2ydydt=y 7. Use the method described in this section to derive the general solution to this system?8E In Exercises 5-12, we consider the partially decoupled system dxdt=2x+2ydydt=y 9. (a) Using the result of Exercise 7. determine the solution that satisfies the initial condition Y(0)=(x(0),y(0))=(0,1) (b) In the xy-phase plane. plot the solution curve associated to this solution. (c) Plot the corresponding x(t)- and y(t)-graphs.In Exercises 5-12, we consider the partially decoupled system dxdt=2x+2ydydt=y 10. (a) Using the result of Exercise 7, determine the solution that satisfies the initial condition Y(0)=(x(0),y(0))=(1,3) (b) In the xy-phase plane, plot the solution curve associated to this solution. (c) Plot the corresponding x(t)- and y(t)-graphs.11E 12E Consider the partially decoupled system dxdt=2x8y2dydt=3y (a) Derive the general solution. (b) Find the equilibrium points of the system. (c)Find the solution that satisfies the initial condition (x0,y0)=(0,1). (d) Use HPGSystemsolver to plot the phase portrait for this system. Identify (he solution curve that corresponds to the solution with initial condition (x0,y0)=(0,1) .1E 2E 3E In Exercises 3—6, a system, an initial condition, a step size, and an integer n are given. The direction field for the system is also provided. (a) Use EulersMethodForSystems to calculate the approximate solution given by Euler’s method for the given system with the given initial condition and step size for n steps. (b) Plot your approximate solution on the direction field. Make sure that your approximate solution is consistent with the direction field. (c) Using HPGSystemSolver, obtain a more detailed sketch of the phase portrait for the system. 5.dxdt=ydydt=sinx{( x 0 , y 0 )=( 0,2)t=0.25n=8 5E 6E Using a computer or calculator, apply Euler’s method to sketch an approximation to the solution curve for the solution to the initial-value problem 2d2ydt2+dydt+4y=0 where (y0, v0) = (2,0). How does your choice of t affect your result?8E Consider the system dxdt=x+ydydt=y (a) Show that the x-axis consists of three solution curves. (b) Using HPGSystemSolver, sketch the solution curves for a number of Initial conditions above and below the x-axis. Do these curve intersect the x-axis? Do they touch the origin? Justify your assertions.2E 3E 4E 5E 6E 7E (a) Suppose Y1(t) is a solution of an autonomous system dY/dt=F(Y)>. Show that Y2(t)=Y1(t+t0) is also a solution for any constant t0. (b) What is the relationship between the solution curves of Y1(t) and Y2(t)?9E Consider the system dxdt=2dydt=y2 (a) Calculate the general solution for the system. (b) What solutions go to infinity? (c)What solutions blow up in finite time?Consider the system dxdt=2dydt=y2 Show that, for the solution (x(t),y(t)) with initial condition (x(0),y(0))=(0,1) , there is a time tsuch that x(t)t* as tt*. In other words, the solution blows up in finite time.[Hint: Note that dy/dt0 for all x and y.]1E In the SIR model, we assume that everyone in the population is susceptible at time t= 0 except the very small fraction that is already infected. Suppose that some fraction of the population has received a vaccine, so they cannot catch the disease. The vaccine makes the fraction of the population that is susceptible at time t = 0 smaller. (a) Using HPGSystemSoIver applied to the SIR model with =0.25 and =0.1, describe the behavior of the solutions with I(0) = 0.01 and S(0) = 0.9, 0.8, 0.7, … Pay particular attention to the maximum of I(t), that is, the maximum number of infecteds for each choice of S(0). Also, note the limit of S(t) as t . (This limit is the fraction of the population that does not catch the disease during the epidemic.) (b) If =0.25 and =0.1, how large a fraction of the population must be vaccinated in order to keep the epidemic from getting started with I(0) = 0.01 ?Vaccines make it possible to prevent epidemics. However, the time it takes to develop a vaccine may make it impossible to vaccinate everyone in a population before a disease arrives. (a) For the SIR model, which initial conditions guarantee that dI/dt 0? [Hint: Your answer should be expressed in terms of the parameters and .] (b) For given values of and , what fraction of a population must be vaccinated before a disease arrives in order to prevent an epidemic?4E 5E One of the basic assumptions of the SIR model is that individuals who recover from the disease never get it again. However, diseases continually evolve, and new strains can emerge that can infect those who have recovered from the previous strain. In this exercise, we modify the SIR model so that recovered become susceptible again in a linear rate. We obtain the system of equations dSdt=SI+RdIdt=SIRdRdt=IR (a) Show that the sum S(t) + I(t) + R(t) is constant as a function of t for this model. (b) Derive a system in the two dependent variables S and I using the fact that R = 1 (S + I) What are the equilibrium points for this model of the two variables S and I? (Hint: Both S and I are nonnegative, and S(t) + I(t) 1 for all t.) (d) Fix =0.3,=0.15, and =0.05 and use HPGSystemSo1ver to sketch the phase portrait. Describe the behavior of solutions. (e) How does the system change if we fix =0.3and=0.15, but vary over a small interval surrounding =0.05?7E 8E 9E Using =1.66 and the value of that you determined in Exercise 9, how would the progress of the epidemic have changed if 200 students had been vaccinated before the disease started? (Give as precise an answer as possible.)1E 2E 3E 4E 5E 1RE Short answer exercises: Exercises 1-14 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers are short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 2. Find all equilibrium points of the system dx/dt=y and dy/dt=ey+x2.Short answer exercises: Exercises 1-14 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers are short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 3. Convert the second-order differential equation d2y/dt2=1 to a first-order system.Short answer exercises: Exercises 1-14 focus on the basic ideas,definitions, and vocabulary of this chapter. Their answers are short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 4. Find the general solution of the system of equations in Exercise 3.Short answer exercises: Exercises 1-14 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers are short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 5. Find all equilibrium points of the system dx/dt=y and dy/dt=sin(xy).Short answer exercises: Exercises 1-14 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers are short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 6. How many equilibrium solutions does the system of differential equations dx/dt=x(xy) and dy/dt=(x24)(y29) have? What are they?7RE 8RE 9RE 10RE 11RE 12RE Short answer exercises: Exercises 1-14 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers are short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 13. Sketch the solution curve for the initial-value problem dx/dt=x,dy/dt=y and (x(0),y(0))=(1,1).14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE In Exercises 31-34, a solution curve in the xy-plane and an initial condition on that curve are specified. Sketch the x(t)- and y(t)- graphs for the solution. 31.32RE 33RE 34RE Consider the partially decoupled system dxdx=x+2y+1dydt=3y (a) Derive the general solution. (b) Find the equilibrium points of the system. (c)Find the solution that satisfies the initial condition (x0,y0)=(1,3). (d) Use HPGSystemsolver to plot the phase portrait for this system. Identify the solution curve that corresponds to the solution with initial condition (x0,y0)=(1,3).Consider the partially decoupled system dxdx=xydydt=y+1 (a) Derive the general solution. (b) Find the equilibrium points of the system. (c)Find the solution that satisfies the initial condition (x0,y0)=(1,0). (d) Use HPGSystemsolver to plot the phase portrait for this system. Identify the solution curve that corresponds to the solution with initial condition (x0,y0)=(1,0).37RE Recall the model dx dt=ax+by dy dt=cx+dy for Paul's and Bob's cafés, where x(t) is Paul's daily profit, y(t) is Bob's daily profit, and $a, b, c,$ and d are parameters governing how the daily profit of each store affects the other. In Exercises 14, different choices of the parameters $a, b, c,$ and d are specified. For each exercise write a brief paragraph describing the interaction between the stores, given the specified parameter values. [For example, suppose a=1,c=1, and b=d=0. If Paul's store is making a profit (x0), then Paul's profit increases more quickly (because ax0 ). However, if Paul makes a profit, then Bob's profits suffer (because cx0 ). since b=d=0, Bob's current profits have no impact on his or Paul's future profits.] 1. a=1,b=1,c=1, and d=1 2E 3E 4E In Exercises 57 , rewrite the specified linear system in matrix form. 5. dxdt=2x+y dydt=x+y In Exercises 57 , rewrite the specified linear system in matrix form. 6. dxdt=3y dydt=3y0.3x In Exercises 57 , rewrite the specified linear system in matrix form. 7. dpdt=3p2q7r dqdt=2p+6r drdt=7.3q+2r 8E In Exercises 89 , rewrite the specified linear system in component form. 9. ( dx dt dy dt )=(0 1)(xy)For the linear systems given in Exercises 1013, use HPGSystemSolver to sketch the direction field, several solutions, and the x(t) - and y(t) -graphs for the solution with initial condition (x,y)=(1,1) 10. dxdt=2x+y dydt=x+y For the linear systems given in Exercises 1013, use HPGSystemsolver to sketch the direction field, several solutions, and the x(t) - and y(t) -graphs for the solution with initial condition (x,y)=(1,1) 11. dxdt=x+2y dydt=xy 12E 13E 14E Let A=(abcd) be a nonzero matrix. That is, suppose that at least one of its entries is nonzero. Show that, if detA=0, then the system dY/dt= AY has an entire line of equilibria. [Hint: First consider the case where a0. Show that any point ( x0,y0 ) that satisfies x0=(b/a)y0 is an equilibrium point. What if we assume that entries of A other than a are nonzero?The general form of a linear, homogeneous, second-order equation with constant coefficientsis d 2 y d t 2 +p dy dt+qy=0 Write the first-order system for this equation, and write this system in matrix form. Show that if q0, then the origin is the only equilibrium point of the system Show that if q0, then the only solution of the second-order equation with y constant is y(t)=0 for all t 17E 18E Convert the third-order differential equation $ d3ydt3+pd2ydt2+qdydt+ry=0 $ where p, q, and r are constants, to a three-dimensional linear system written in matrix form. In Exercises 2023, we consider the following model of the market for single-family housing in a community. Let S(t) be the number of sellers at time t, and let B(t) be the number of buyers at time t We assume that there are natural equilibrium levels of buyers and sellers (made up of people who retire, change job locations, or wish to move for family reasons). The equilibrium level of sellers is S0 and the equilibrium level of buyers is B0 However, market forces can entice people to buy or sell under various conditions. For example, if the price of a house is very high, then house owners are tempted to sell their homes. If prices are very low, extra buyers enter the market looking for bargains. We let b(t)=B(t)B0 denote the deviation of the number of buyers from equilibrium at time t. So if b(t)0, then there are more buyers than usual, and we say it is a "seller's market." Presumably the competition of the extra buyers for the same number of houses for sale will force the prices up (the law of supply and demand). Similarly, we let s(t)=S(t)S0 denote the deviation of the number of sellers from the equilibrium level. If s(t)0 , then there are more sellers on the market than usual; and if the number of buyers is low, there are too many houses on the market and prices decrease, which in turn affects decisions to buy or sell. We can give a simple model of this situation as follows: dYdt=AY=()(bs),whereY=(bs) The exact values of the parameters ,,, and depend on the economy of a particular community. Nevertheless, if we assume that everybody wants to get a bargain when they are buying a house and to get top dollar when they are selling a house, then we can hope to predict whether the parameters are positive or negative even though we cannot predict their exact values. Use the information given above to obtain information about the parameters , , and . Be sure to justify your answers.20E 21E 22E 23E Consider the linear system dYdt=(2011)Y Show that the two functions Y1(t)=(0 e t )andY2(t)=( e 2t e 2t ) are solutions to the differential equation. Solve the initial-value problem dYdt=(2011)Y,Y(0)=( 2 1)Consider the linear system dYdt=(1 113)Y (a)Show that the function Y(t)=( t e 2t (t+1) e 2t ) is a solution to the differential equation. (b)Solve the initial-value problem dYdt=(1 113)Y,Y(0)=(02) In Exercises 2629, a coefficient matrix for the linear system dYdt=AY,whereY(t)=( x(t) y(t)) is specified. Also two functions and an initial value are given. For each system: Check that the two functions are solutions of the system; if they are not solutions, then stop. Check that the two solutions are linearly independent; if they are not linearly independent, then stop. Find the solution to the linear system with the given initial value.26E 27E A=( 2 33 2) Functions: Y1(t)=e2t(cos3t,sin3t) Y2(t)=e2t(sin3t,cos3t) Initial value: Y(0)=(2,3)29E 30E 31E 32E 33E 34E 35E In Exercises 110 (a) compute the eigenvalues; (b) for each eigenvalue, compute the associated eigenvectors; (c) using HPGSystemsolver, sketch the direction field for the system, and plot the straight-line solutions; (d) for each eigenvalue, specify a corresponding straight-line solution and plot its x(t) and y(t) -graphs; and (e) if the system has two distinct eigenvalues, compute the general solution. 1. dYdt=(320 2)Y In Exercises 110 (a) compute the eigenvalues; (b) for each eigenvalue, compute the associated eigenvectors; (c) using HPGSystemSol ver, sketch the direction field for the system, and plot the straight-line solutions; (d) for each eigenvalue, specify a corresponding straight-line solution and plot its x(t) and y(t) -graphs; and (e) if the system has two distinct eigenvalues, compute the general solution. 2. dYdt=( 4 2 1 3)Y In Exercises 110 (a) compute the eigenvalues; (b) for each eigenvalue, compute the associated eigenvectors; (c) using HPGSystemSolver, sketch the direction field for the system, and plot the straight-line solutions; (d) for each eigenvalue, specify a corresponding straight-line solution and plot its x(t) and y(t) -graphs; and (e) if the system has two distinct eigenvalues, compute the general solution. 3. ( dx dt dy dt )=( 5 2 1 4)(xy)In Exercises 110 (a) compute the eigenvalues; (b) for each eigenvalue, compute the associated eigenvectors; (c) using HPGSystemSol ver, sketch the direction field for the system, and plot the straight-line solutions; (d) for each eigenvalue, specify a corresponding straight-line solution and plot its x(t) and y(t) -graphs; and (e) if the system has two distinct eigenvalues, compute the general solution. 4. ( dx dt dy dt )=(21 14)(xy)5E In Exercises 110 (a) compute the eigenvalues; (b) for each eigenvalue, compute the associated eigenvectors; (c) using HPGSystemSolver, sketch the direction field for the system, and plot the straight-line solutions; (d) for each eigenvalue, specify a corresponding straight-line solution and plot its x(t) and y(t) -graphs; and (e) if the system has two distinct eigenvalues, compute the general solution. 6. dxdt=5x+4y dydt=9x In Exercises 110 (a) compute the eigenvalues; (b) for each eigenvalue, compute the associated eigenvectors; (c) using HPGSystemsolver, sketch the direction field for the system, and plot the straight-line solutions; (d) for each eigenvalue, specify a corresponding straight-line solution and plot its x(t) and y(t) -graphs; and (e) if the system has two distinct eigenvalues, compute the general solution. 7. ( dx dt dy dt )=(3410)(xy)8E In Exercises 110 (a) compute the eigenvalues; (b) for each eigenvalue, compute the associated eigenvectors; (c) using HPGSystemSol ver, sketch the direction field for the system, and plot the straight-line solutions; (d) for each eigenvalue, specify a corresponding straight-line solution and plot its x(t) and y(t) -graphs; and (e) if the system has two distinct eigenvalues, compute the general solution. 9. dxdt=2x+y dydt=x+y In Exercises $1-10$ (a) compute the eigenvalues; (b) for each eigenvalue, compute the associated eigenvectors; (c) using HPGSystemSolver, sketch the direction field for the system, and plot the straight-line solutions; (d) for each eigenvalue, specify a corresponding straight-line solution and plot its x(t) and y(t) -graphs; and (e) if the system has two distinct eigenvalues, compute the general solution. 10. dxdt=x2y dydt=x4y Solve the initial-value problem dx dt=2x2y dy dt=2x+y where the initial condition (x(0),y(0)) is: (a) (1,0) (b) (0,1) (c) (1,2)12E Solve the initial-value problem dYdt=( 412 3)Y,Y(0)=Y0 where the initial condition Y0 is: (a) Y0=(1,0) (b) Y0=(2,1)(c) Y0=(1,2)14E Show that a is the only eigenvalue and that every nonzero vector is an eigenvector for the matrix A=(a00a)A matrix of the form A=(ab0d) is called upper triangular. Suppose that b0 and ad. Find the eigenvalues and eigenvectors of A .A matrix of the form B=(abbd) is called symmetric. Show that B has real eigenvalues and that, if b0, then B has two distinct eigenvalues.18E Consider the second-order equation d2ydt2+pdydt+qy=0 where p and q are positive. (a) Convert this equation into a first-order, linear system. (b) Compute the characteristic polynomial of the system. (c) Find the eigenvalues. (d) Under what conditions on p and q are the eigenvalues two distinct real num- bers? (e) Verify that the eigenvalues are negative if they are real numbers.For the harmonic oscillator with mass m=1, spring constant k=4, and damping coefficient b=5 (a) compute the eigenvalues and associated eigenvectors; (b) for each eigenvalue, pick an associated eigenvector V and determine the solution Y(t) with Y(0)=V (c) for each solution derived in part (b), plot its solution curve in the y v-phase plane; (d) for each solution derived in part (b), plot its y(t) - and v(t) -graphs; and (e) for each solution derived in part (b), give a brief description of the behavior of the mass-spring system.21E In Exercises 21-24, we return to Exercises 1-4 in Section 2.3 . (For convenience, the equations are reproduced below.) For each second-order equation, (a) convert the equation to a first-order, linear system; (b) compute the eigenvalues and eigenvectors of the system; (c) for each eigenvalue, pick an associated eigenvector V, and determine the solution Y(t) to the system; and (d) compare the results of your calculations in part (c) with the results that you obtained when you used the guess-and-test method of Section 2.3 22. d2ydt2+5dydt+6y=0 23E 24E 25E In Exercises 18, we refer to linear systems from the exercises in Section 3.2 . Sketch the phase portrait for the system specified. 1.The system in Exercise 1, Section 3.2 In Exercises 18, we refer to linear systems from the exercises in Section 3.2 . Sketch the phase portrait for the system specified. 2. The system in Exercise 2, Section 3.2 3E In Exercises 18, we refer to linear systems from the exercises in Section 3.2 . Sketch the phase portrait for the system specified. 4. The system in Exercise 6, Section 3.2 5E 6E In Exercises 1-8, we refer to linear systems from the exercises in Section 3.2 . Sketch the phase portrait for the system specified. 7. The system in Exercise 9, Section 3.2 8E In Exercises 912, we refer to initial-value problems from the exercises in Section 3.2 Sketch the solution curves in the phase plane and the x(t) - and y(t) -graphs for the solutions corresponding to the initial-value problems specified. 9. The initial-value problems in Exercise 11, Section 3.2 10E 11E 12E In Exercises 13-16, we refer to the second-order equations from the exercises in Section 3.2. Sketch the phase portrait for the second-order equations specified. 13. The second-order equation in Exercise 21 , Section 3.2 14E 15E 16E 17E 18E The slope field for the system dx dt=2x+12y dy dt=y is shown to the right. (a) Determine the type of the equilibrium point at the origin. (b) Calculate all straight-line solutions. (c) Plot the x(t) - and y(t) -graphs, (t0) for the initial conditions A=(2,1) B=(1,2),C=(2,2), and D= (2,0)20E 21E 22E 23E 24E 25E 26E Consider the linear system dYdt=( 2102)Y $ (a) Show that (0,0) is a saddle. (b) Find the eigenvalues and eigenvectors and sketch the phase plane. (c) On the phase plane, sketch the solution curves with initial conditions (1,0.01) and (1,-0.01) (d) Estimate the time t at which the solutions with initial conditions (1,0.01) and(1,-0.01) will be 1 unit apart.Suppose that the 22 matrix A has =1+3i as an eigenvalue with eigenvector Y0=( 2+i1) Compute the general solution to dY/dt=AY Suppose that the 22 matrix B has =2+5i as an eigenvalue with eigenvector Y0=(1 43i) Compute the general solution to dY/dt=BY In Exercises 3-8, each linear system has complex eigenvalues. For each system, (a) find the eigenvalues; (b) determine if the origin is a spiral sink, a spiral source, or a center; (c) determine the natural period and natural frequency of the oscillations, (d) determine the direction of the oscillations in the phase plane (do the solutions goclockwise or counterclockwise around the origin?); and (e) using HPGSystemsolver, sketch the x y -phase portrait and the x(t) - and y(t) graphs for the solutions with the indicated initial conditions. 3. dYdt=(02 20)Y, with initial condition Y0=(1,0)In Exercises 3-8, each linear system has complex eigenvalues. For each system, (a) find the eigenvalues; (b) determine if the origin is a spiral sink, a spiral source, or a center; (c) determine the natural period and natural frequency of the oscillations, (d) determine the direction of the oscillations in the phase plane (do the solutions go clockwise or counterclockwise around the origin?); and (e) using HPGSystemSolver, sketch the x y -phase portrait and the x(t) - and y(t) graphs for the solutions with the indicated initial conditions. 4. dYdt=(22 46)Y, with initial condition Y0=(1,1) .In Exercises 3-8, each linear system has complex eigenvalues. For each system, (a) find the eigenvalues; (b) determine if the origin is a spiral sink, a spiral source, or a center; (c) determine the natural period and natural frequency of the oscillations, (d) determine the direction of the oscillations in the phase plane (do the solutions go clockwise or counterclockwise around the origin?); and (e) using HPGSystemsolver, sketch the x y -phase portrait and the x(t) - and y(t) graphs for the solutions with the indicated initial conditions. 5. dYdt=( 3 531)Y, with initial condition Y0=(4,0)In Exercises 3-8, each linear system has complex eigenvalues. For each system, (a) find the eigenvalues; (b) determine if the origin is a spiral sink, a spiral source, or a center; (c) determine the natural period and natural frequency of the oscillations, (d) determine the direction of the oscillations in the phase plane (do the solutions go clockwise or counterclockwise around the origin?); and (e) using HPGSystemSolver, sketch the x y -phase portrait and the x(t) - and y(t) graphs for the solutions with the indicated initial conditions. 6. dYdt=(02 2 1)Y, with initial condition Y0=(1,1)In Exercises 3-8, each linear system has complex eigenvalues. For each system, (a) find the eigenvalues; (b) determine if the origin is a spiral sink, a spiral source, or a center; (c) determine the natural period and natural frequency of the oscillations, (d) determine the direction of the oscillations in the phase plane (do the solutions go clockwise or counterclockwise around the origin?); and (e) using HPGSystemSolver, sketch the x y -phase portrait and the x(t) - and y(t) graphs for the solutions with the indicated initial conditions. 7. dYdt=(2 621)Y, with initial condition Y0=(2,1)8E In Exercises 9-14, the linear systems are the same as in Exercises 3-8. Forleach system, (a) find the general solution; (b) find the particular solution with the given initial value; and (c) sketch the x(t) - and y(t) -graphs of the particular solution. (Compare these sketches with the sketches you obtained in the corresponding problem from Exercises 38. ) 9. dYdt=(02 20)Y, with initial condition Y0=(1,0)In Exercises 9-14, the linear systems are the same as in Exercises 38 . Forleach system, (a) find the general solution; (b) find the particular solution with the given initial value; and (c) sketch the x(t) - and y(t) -graphs of the particular solution. (Compare these sketches with the sketches you obtained in the corresponding problem from Exercises 38. ) 10. dYdt=(22 46)Y, with initial condition Y0=(1,1)11E 12E In Exercises 9-14, the linear systems are the same as in Exercises 38 . Forleach system, (a) find the general solution; (b) find the particular solution with the given initial value; and (c) sketch the x(t) - and y(t) -graphs of the particular solution. (Compare these sketches with the sketches you obtained in the corresponding problem from Exercises 38. ) 13. dYdt=(2 621)Y, with initial condition Y0=(2,1)14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E In Exercises 1-4, each of the linear systems has one eigenvalue and one line of eigenvectors. For each system, (a) find the eigenvalue; (b) find an eigenvector; (c) sketch the direction field; (d) sketch the phase portrait, including the solution curve with initial condition Y0=(1,0); and (e) sketch the x(t) - and y(t) -graphs of the solution with initial condition Y0=(1,0) 1. dYdt=( 301 3)Y 2E 3E 4E In Exercises 5-8, the linear systems are the same as those in Exercises 1-4 . For each system, (a) find the general solution; (b) find the particular solution for the initial condition Y0=(1,0); and (c) sketch the x(t) - and y(t) -graphs of the solution. (Compare these sketches with the sketches you obtained in the corresponding problem from Exercises 14. ) 5. dYdt=( 301 3)Y 6E 7E 8E Given a quadratic 2++, what condition on and guarantees (a) that the quadratic has a double root? (b) that the quadratic has zero as a root?10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E

Page: [1][2][3][4][5]