Differential Equations
4th Edition
ISBN: 9781133383093
Author: Paul Blanchard; Robert L. Devaney; Glen R. Hall
Publisher: Cengage Learning US
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Chapter 1, Problem 11RE
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Chapter 1 Solutions
Differential Equations
Ch. 1.1 - In Exercises 1 and 2, find the equilibrium...Ch. 1.1 - In Exercises 1 and 2, find the equilibrium...Ch. 1.1 - Consider the population model dPdt=0.4P(1P230)...Ch. 1.1 - Consider the population model ...Ch. 1.1 - Consider the differential equation dydt=y3y212y...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...
Ch. 1.1 - MacQuarie Island is a small island about half-way...Ch. 1.1 - The velocity u of a freefalling skydiver is well...Ch. 1.1 - Exercises 13—15 consider an elementary model of...Ch. 1.1 - Exercises 13—15 consider an elementary model of...Ch. 1.1 - Exercises 13—15 consider an elementary model of...Ch. 1.1 - The expenditure on education in the U.S. is given...Ch. 1.1 - Suppose a species of fish in a particular lake has...Ch. 1.1 - Suppose that the growth-rate parameter k = 0.3 and...Ch. 1.1 - The rhinoceros is now extremely rare. Suppose...Ch. 1.1 - While it is difficult to imagine a time before...Ch. 1.1 - For the following predator-prey systems, identify...Ch. 1.1 - In the following predator-prey population models,x...Ch. 1.1 - The following systems are models of the...Ch. 1.2 - Bob. Glen. and Paul are once again sitting around...Ch. 1.2 - Make up a differential equation of the form ...Ch. 1.2 - Make up a differential equation of the form dy/dt...Ch. 1.2 - In Section 1.1, we guessed solutions to the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 524, find the general solution of the...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - Prob. 32ECh. 1.2 - Prob. 33ECh. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - In Exercises 2538, solve the given initial-value...Ch. 1.2 - Prob. 38ECh. 1.2 - A 5-gallon bucket is full of pure water. Suppose...Ch. 1.2 - Consider the following very simple model of blood...Ch. 1.2 - A cup of hot chocolate is initially 170o Fand is...Ch. 1.2 - Suppose you are having a dinner party for a large...Ch. 1.2 - Prob. 43ECh. 1.3 - In Exercises 1-6, sketch the slope fields for the...Ch. 1.3 - Prob. 2ECh. 1.3 - Prob. 3ECh. 1.3 - Prob. 4ECh. 1.3 - Prob. 5ECh. 1.3 - Prob. 6ECh. 1.3 - In Exercises 710, a differential equation and its...Ch. 1.3 - Prob. 8ECh. 1.3 - Prob. 9ECh. 1.3 - Prob. 10ECh. 1.3 - Suppose we know that the function f(t, y) is...Ch. 1.3 - Prob. 12ECh. 1.3 - Prob. 13ECh. 1.3 - Prob. 14ECh. 1.3 - Consider the autonomous differential equation ...Ch. 1.3 - Eight differential equations and four slope fields...Ch. 1.3 - Prob. 17ECh. 1.3 - Prob. 18ECh. 1.3 - Prob. 19ECh. 1.3 - Prob. 20ECh. 1.3 - Prob. 21ECh. 1.3 - Prob. 22ECh. 1.4 - In Exercises 14, use EulersMethod to perform...Ch. 1.4 - In Exercises 14, use EulersMethod to perform...Ch. 1.4 - Prob. 3ECh. 1.4 - In Exercises 14, use EulersMethod to perform...Ch. 1.4 - In Exercises 510, use Euler’s method with the...Ch. 1.4 - In Exercises 510, use Euler’s method with the...Ch. 1.4 - Prob. 7ECh. 1.4 - Prob. 8ECh. 1.4 - Prob. 9ECh. 1.4 - Prob. 10ECh. 1.4 - Prob. 11ECh. 1.4 - Prob. 12ECh. 1.4 - Prob. 13ECh. 1.4 - Prob. 14ECh. 1.4 - Consider the initial-value problem dy/dt= y ,y(0)...Ch. 1.4 - Consider the initial-value problem dy/dt= 2 ...Ch. 1.4 - As we saw in Exercise 19 of Section 1.3, the...Ch. 1.4 - Prob. 18ECh. 1.4 - Prob. 19ECh. 1.4 - Prob. 20ECh. 1.4 - Prob. 21ECh. 1.5 - In Exercises 1—4, we refer to a function f, but we...Ch. 1.5 - Prob. 2ECh. 1.5 - In Exercises 1—4, we refer to a function f, but we...Ch. 1.5 - In Exercises 1—4, we refer to a function f, but we...Ch. 1.5 - Prob. 5ECh. 1.5 - In Exercises 5—8, an initial condition for the...Ch. 1.5 - Prob. 7ECh. 1.5 - In Exercises 5—8, an initial condition for the...Ch. 1.5 - (a) Show that y1(t)=t2 and y2(t)=t2+1 are...Ch. 1.5 - Consider the differential equation dy/dt=2y (a)...Ch. 1.5 - Consider the differential equation dydt=yt2 (a)...Ch. 1.5 - (a) Show that y1(t)=1t1 and y2(t)=1t2 are...Ch. 1.5 - Prob. 13ECh. 1.5 - In Exercises 13—16, an initial-value problem is...Ch. 1.5 - In Exercises 13—16, an initial-value problem is...Ch. 1.5 - In Exercises 13—16, an initial-value problem is...Ch. 1.5 - Prob. 17ECh. 1.5 - We have emphasized that the Uniqueness Theorem...Ch. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - Prob. 3ECh. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - Prob. 5ECh. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - Prob. 9ECh. 1.6 - In Exercises 112, sketch the phase lines for the...Ch. 1.6 - Prob. 11ECh. 1.6 - Prob. 12ECh. 1.6 - In Exercises 1321, a differential equation and...Ch. 1.6 - In Exercises 1321, a differential equation and...Ch. 1.6 - In Exercises 1321, a differential equation and...Ch. 1.6 - Prob. 16ECh. 1.6 - ]In Exercises 1321, a differential equation and...Ch. 1.6 - Prob. 18ECh. 1.6 - In Exercises 1321, a differential equation and...Ch. 1.6 - Prob. 20ECh. 1.6 - Prob. 21ECh. 1.6 - In Exercises 2227, describe the long-term behavior...Ch. 1.6 - Prob. 23ECh. 1.6 - In Exercises 2227, describe the long-term behavior...Ch. 1.6 - In Exercises 2227, describe the long-term behavior...Ch. 1.6 - In Exercises 2227, describe the long-term behavior...Ch. 1.6 - Prob. 27ECh. 1.6 - Prob. 28ECh. 1.6 - Prob. 29ECh. 1.6 - In Exercises 2932, the graph of a function f(y) is...Ch. 1.6 - In Exercises 2932, the graph of a function f(y) is...Ch. 1.6 - In Exercises 2932, the graph of a function f(y) is...Ch. 1.6 - Prob. 33ECh. 1.6 - Prob. 34ECh. 1.6 - Prob. 35ECh. 1.6 - Prob. 36ECh. 1.6 - Eight differential equations and four phase lines...Ch. 1.6 - Prob. 38ECh. 1.6 - Prob. 39ECh. 1.6 - Consider the Ermentrout-Kopell model for the...Ch. 1.6 - Prob. 41ECh. 1.6 - Prob. 42ECh. 1.6 - Prob. 43ECh. 1.6 - Prob. 44ECh. 1.6 - Let x(t) be the amount of time between two...Ch. 1.6 - Prob. 46ECh. 1.6 - Use the model in Exercise 45 to predict what...Ch. 1.6 - Prob. 48ECh. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - In Exercises 1-6, locate the bifurcation values...Ch. 1.7 - Prob. 6ECh. 1.7 - Prob. 7ECh. 1.7 - Prob. 8ECh. 1.7 - Prob. 9ECh. 1.7 - Prob. 10ECh. 1.7 - The graph to the right is the graph of a function...Ch. 1.7 - The graph to the right is the graph of a function...Ch. 1.7 - Six one-parameter families of different equations...Ch. 1.7 - Consider the Ermentrout-Kopell model for the...Ch. 1.7 - Prob. 15ECh. 1.7 - Sketch the graph of a function g(y) such that the...Ch. 1.7 - Is it possible to find a continuous function...Ch. 1.7 - Prob. 18ECh. 1.7 - Consider the population model dPdt=2PP250 for a...Ch. 1.7 - Prob. 20ECh. 1.7 - Prob. 21ECh. 1.7 - Prob. 22ECh. 1.7 - Prob. 23ECh. 1.8 - In Exercises 1-6, find the general solution of the...Ch. 1.8 - Prob. 2ECh. 1.8 - In Exercises 1-6, find the general solution of the...Ch. 1.8 - Prob. 4ECh. 1.8 - In Exercises 1-6, find the general solution of the...Ch. 1.8 - In Exercises 1-6, find the general solution of the...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - In Exercises 7-12, solve the given initial-value...Ch. 1.8 - Consider the nonhomogeneous linear equation...Ch. 1.8 - Prob. 14ECh. 1.8 - Prob. 15ECh. 1.8 - Prob. 16ECh. 1.8 - Consider the nonlinear differential equation...Ch. 1.8 - Prob. 18ECh. 1.8 - Prob. 19ECh. 1.8 - Consider the nonhomogeneous linear equation...Ch. 1.8 - In Exercises 21-24, find the general solution and...Ch. 1.8 - In Exercises 21-24, find the general solution and...Ch. 1.8 - In Exercises 21-24, find the general solution and...Ch. 1.8 - In Exercises 21-24, find the general solution and...Ch. 1.8 - Prob. 25ECh. 1.8 - Prob. 26ECh. 1.8 - In Exercises 25-28, give a brief qualitative...Ch. 1.8 - Prob. 28ECh. 1.8 - A person initially places $1,000 in a savings...Ch. 1.8 - A student has saved $70,000 for her college...Ch. 1.8 - A college professor contributes $5,000 per year...Ch. 1.8 - Prob. 32ECh. 1.8 - Prob. 33ECh. 1.8 - Prob. 34ECh. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 1-6, find the general solution of the...Ch. 1.9 - In Exercises 7-12, solve the given initial-value...Ch. 1.9 - In Exercises 7-12, solve the given initial-value...Ch. 1.9 - Prob. 9ECh. 1.9 - Prob. 10ECh. 1.9 - Prob. 11ECh. 1.9 - In Exercises 7-12, solve the given initial-value...Ch. 1.9 - In Exercises 13-18, the differential equation is...Ch. 1.9 - Prob. 14ECh. 1.9 - Prob. 15ECh. 1.9 - Prob. 16ECh. 1.9 - Prob. 17ECh. 1.9 - Prob. 18ECh. 1.9 - Prob. 19ECh. 1.9 - Prob. 20ECh. 1.9 - Prob. 21ECh. 1.9 - Prob. 22ECh. 1.9 - Prob. 23ECh. 1.9 - A 30-gallon tank initially contains 15 gallons of...Ch. 1.9 - A 400-gallon tank initially contains 200 gallons...Ch. 1.9 - A 100-gallon tank initially contains loo gallons...Ch. 1.9 - Suppose a 50-gallon tank contains a volume Vo of...Ch. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Prob. 4RECh. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Short answer exercises: Exercises 1-10 focus on...Ch. 1 - Prob. 7RECh. 1 - Prob. 8RECh. 1 - Prob. 9RECh. 1 - Prob. 10RECh. 1 - Prob. 11RECh. 1 - Prob. 12RECh. 1 - Prob. 13RECh. 1 - Prob. 14RECh. 1 - True-false: For Exercises 11-20, determine if the...Ch. 1 - Prob. 16RECh. 1 - True-false: For Exercises 11-20, determine if the...Ch. 1 - Prob. 18RECh. 1 - Prob. 19RECh. 1 - Prob. 20RECh. 1 - Prob. 21RECh. 1 - Prob. 22RECh. 1 - Prob. 23RECh. 1 - Prob. 24RECh. 1 - Prob. 25RECh. 1 - Prob. 26RECh. 1 - Prob. 27RECh. 1 - Prob. 28RECh. 1 - In Exercises 21-29, (a) specify if the given...Ch. 1 - Prob. 30RECh. 1 - Prob. 31RECh. 1 - Prob. 32RECh. 1 - Prob. 33RECh. 1 - Prob. 34RECh. 1 - In Exercises 30-39, (a) specify if the given...Ch. 1 - Prob. 36RECh. 1 - Prob. 37RECh. 1 - Prob. 38RECh. 1 - Prob. 39RECh. 1 - Prob. 40RECh. 1 - Prob. 41RECh. 1 - Consider the autonomous differential equation...Ch. 1 - Prob. 43RECh. 1 - Prob. 44RECh. 1 - Prob. 45RECh. 1 - Prob. 46RECh. 1 - Prob. 47RECh. 1 - Prob. 48RECh. 1 - Eight differential equations and four slope fields...Ch. 1 - Prob. 50RECh. 1 - Prob. 51RECh. 1 - Consider the differential equation dy/dt=2ty2 ....Ch. 1 - Prob. 53RECh. 1 - A 1000-galIon tank initially contains a mixture of...
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- 12. For the differential equation in the previous question, what is the correct form for a particularsolution?A. yp = Ae^t + Bt^2 B. yp = Ae^t + Bt^2 + Ct + DC. yp = Ate^t + Bt^2 D. yp = Ate^t + Bt^2 + Ct + D Previous differential equation y′′ − 4y′ + 3y = e^t + t^2arrow_forward16. The appropriate form for the particular solution yp(x) of y^(3) − y′′ − 2y′ = x^2 + e^2x isA. yp(x) = Ax^2 + Bx + C + De^2x B. yp(x) = Ax^3 + Bx^2 + Cx + Dxe^2xC. yp(x) = Ax^2 +Be^2x D. yp(x) = A+Be^2x +Ce^−x E. yp(x) = Ax^2 +Bx+C +(Dx+E)e^2xarrow_forwardDistance Between Two Ships Two ships leave the same port at noon. Ship A sails north at 17 mph, and ship B sails east at 11 mph. How fast is the distance between them changing at 1 p.m.? (Round your answer to one decimal place.) 20.3 X mph Need Help? Read It Watch It SUBMIT ANSWERarrow_forward
- 11. Consider the 2nd-order non-homogeneous differential equation y′′ − 4y′ + 3y = et + t2What is the complementary (or homogeneous) solution?A. yc = c1e^t + c2t^2 B. yc = c1e^−t + c2e^−3t C. yc = c1e^t + c2e^3t D. yc = c1e^t + c2e^−3tarrow_forward5. A trial solution for the non-homogeneous equation y′′ + y′ − 2y = e^x isA. Ae^x B. Ae^x+ Be^−2x C. Ae^x + Be^−x D. Axe^x E. None of these.arrow_forward14. Write u = - sint-cost in the form u = C cos(t - a) with C > 0 and 0 ? PAUSE Z X C VI B N Marrow_forward
- 19. If the method of undetermined coefficients is used, the form of a particular solution ofy^(4) − y = e^−t + 3 sin(t) isA. yp(t) = Ate^−t + B cos(t) + C sin(t)B. yp(t) = At^2e^−t + B cos(t) + C sin(t)C. yp(t) = Ate^−t + Bt cos(t) + Ct sin(t)D. yp(t) = At^2e^−t + Bt cos(t) + Ct sin(t)E. yp(t) = Ate^−t + Bt sin(t)arrow_forward15. A spring-mass system is governed by the differential equation 2x′′ + 72x = 100 sin(3ωt) .For what value of ω will resonance occur?A. 3 B. 6√2 C. 2 D. 10 E. No valuearrow_forwardQuestion 3. A manufacturer has modeled its yearly production function P (the value of its entire production, in millions of dollars) as a Cobb-Douglas function P(L, K) = 1.47L0.65 0.35 where L is the number of labor hours (in thousands) and K is the invested capital (in millions of dollars). ӘР Ət (a) Express the rate of change of production 07-2 in time, in terms of the rate of change of the labor force and the rate of change of the capital in time. (b) Suppose that when L = 30 and K = 8, the labor force is decreasing at a rate of 2000 labor hours per year and capital is increasing at a rate of 500,000 per year. What is the rate of change of production per year?arrow_forward
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