DIFFERENTIAL EQUATIONS-NEXTGEN WILEYPLUS
3rd Edition
ISBN: 9781119764564
Author: BRANNAN
Publisher: WILEY
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Chapter 1.2, Problem 8P
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For the following problems:
ii.
Identify the equilibrium values
Construct a phase line. Identify the signs of y' and y".
Sketch several solution curves.
a. = (y + 2)(y - 3)
dx
b. = y²-2y
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H3.
Chapter 1 Solutions
DIFFERENTIAL EQUATIONS-NEXTGEN WILEYPLUS
Ch. 1.1 - Newton’s Law of Cooling. A cup of hot coffee has...Ch. 1.1 - Blood plasma is stored at . Before it can be...Ch. 1.1 - At 11:09p.m. a forensics expert arrives at a crime...Ch. 1.1 - The rate constant if the population doubles in ...Ch. 1.1 - The field mouse population in Example 3 satisfies...Ch. 1.1 - Radioactive Decay. Experiments show that a...Ch. 1.1 - A radioactive material, such as the isotope...Ch. 1.1 - Classical Mechanics. The differential equation for...Ch. 1.1 - For small, slowly falling objects, the assumption...Ch. 1.1 - Mixing Problems. Many physical systems can be cast...
Ch. 1.1 - Mixing Problems. Many physical systems can be cast...Ch. 1.1 - Mixing Problems. Many physical systems can be cast...Ch. 1.1 - Pharmacokinetics. A simple model for the...Ch. 1.1 - A certain drug is being administered intravenously...Ch. 1.1 - Continuously Compounded Interest. The amount of...Ch. 1.1 - Continuously Compounded Interest. The amount of...Ch. 1.1 - Continuously Compounded Interest. The amount of...Ch. 1.1 - Continuously Compounded Interest. The amount of...Ch. 1.1 - A spherical raindrop evaporates at a rate...Ch. 1.1 - Prob. 20PCh. 1.2 - Phase Line Diagrams. Problems through involve...Ch. 1.2 - Phase Line Diagrams. Problems 1 through 7 involve...Ch. 1.2 - Phase Line Diagrams. Problems through involve...Ch. 1.2 - Phase Line Diagrams. Problems 1 through 7 involve...Ch. 1.2 - Phase Line Diagrams. Problems through involve...Ch. 1.2 - Phase Line Diagrams. Problems 1 through 7 involve...Ch. 1.2 - Phase Line Diagrams. Problems through involve...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems through involve equations of the form ....Ch. 1.2 - Direction Fields. In each of problems 14 through...Ch. 1.2 - Direction Fields. In each of problems through...Ch. 1.2 - Direction Fields. In each of problems 14 through...Ch. 1.2 - Direction Fields. In each of problems through...Ch. 1.2 - Direction Fields. In each of problems 14 through...Ch. 1.2 - Direction Fields. In each of problems through...Ch. 1.2 - In each of problems through draw a direction...Ch. 1.2 - In each of problems 20 through 23 draw a direction...Ch. 1.2 - In each of problems through draw a direction...Ch. 1.2 - In each of problems through draw a direction...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Verify that the function in Eq.(11) is a solution...Ch. 1.2 - Show that Asint+Bcost=Rsin(t), where R=A2+B2 and ...Ch. 1.2 - If in the exponential model for population growth,...Ch. 1.2 - An equation that is frequently used to model the...Ch. 1.2 - In addition to the Gompertz equation (see Problem...Ch. 1.2 - A chemical of fixed concentration flows into a...Ch. 1.2 - A pond forms as water collects in a conical...Ch. 1.2 - The Solow model of economic growth (ignoring the...Ch. 1.3 - In each of Problems through , determine the order...Ch. 1.3 - In each of Problems 1 through 6, determine the...Ch. 1.3 - In each of Problems 1 through 6, determine the...Ch. 1.3 - In each of Problems through , determine the order...Ch. 1.3 - In each of Problems through , determine the order...Ch. 1.3 - In each of Problems through , determine the order...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - In each of Problems 13 through 20, verify that...Ch. 1.3 - In each of Problems through , verify that each...Ch. 1.3 - In each of Problems through , verify that each...Ch. 1.3 - In each of Problems 13 through 20, verify that...Ch. 1.3 - In each of Problems 13 through 20, verify that...Ch. 1.3 - In each of Problems through , verify that each...Ch. 1.3 - In each of Problems 13 through 20, verify that...Ch. 1.3 - In each of Problems through , verify that each...Ch. 1.3 - In each of Problems 21 through 24, determine the...Ch. 1.3 - In each of Problems through , determine the...Ch. 1.3 - In each of Problems through , determine the...Ch. 1.3 - In each of Problems 21 through 24, determine the...Ch. 1.3 - In each of Problems 25 and 26, determine the...Ch. 1.3 - In each of Problems 25 and 26, determine the...Ch. 1.3 - In Problems 27 through 31, verify that y(t)...Ch. 1.3 - In Problems through , verify that satisfies the...Ch. 1.3 - In Problems through , verify that satisfies the...Ch. 1.3 - In Problems 27 through 31, verify that y(t)...Ch. 1.3 - In Problems through , verify that satisfies the...Ch. 1.3 - Verify that the function (t)=c1et+c2e2t is a...Ch. 1.3 - Verify that the function is a solution of the...Ch. 1.3 - Verify that the function (t)=c1etcos2t+c2etsin2t...
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- [6] An equation dt = f(y) has the following phase portrait. 2 Y (a) Find all equilibrium solutions. (b) Determine whether each of the equilibrium solutions is stable, asymptotically stable or unstable. (c) Graph the solutions y(t) vs t, for the initial values y(1.4) = 0, y(0) = 0.5, y(0) = 1, y (0) = 1.1, y(0) = 1.5, y(-0.5) = 1.5, y(0) = 2, y(0) = 2.5, y(0) = 3, y(0) = 3.5, y(0) = 4, y(0) = 4.5, y(-1) = 4.5. (Without further quantitative information about the equation and the solution formula, it's clearly impossible to draw accurate graphs of y(t) vs t. Here, try to sketch graphs qualitatively to show the correct dynamic properties. The point is that a great deal of info about solution dynamics can be read off from one simple figure of phase portrait.)arrow_forward5arrow_forwardPlease solve & show steps...arrow_forward
- 7) In each of the following problems:a. Sketch the Phase Plot of the ODE.b. Determine the equilibrium solutions.c. Classify the equilibrium solutions.d. Draw the phase line and sketch several graphs of solutions on the ty-plane. (7a) y′ = y(y −1)(y −2) , y0 > 0 (7b) y′ = y (1 −y2) , −∞< y0 < ∞. (7c) y′ = y2(1 −y)2, −∞< y0 < ∞. carrow_forwardIn each of the following problems, sketch the graph of f(y) versus y, determine the equilibrium solutions, and classify each one as asymptotically stable, asymptotically unstable, or semi-stable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. Here y0 = y(0)arrow_forwardConsider the equation (31) dydt=ay−y2=y(a−y) a.Again consider the cases a < 0, a = 0, and a > 0. In each case find the critical points, draw the phase line, and determine whether each critical point is asymptotically stable, semistable, or unstable.arrow_forward
- Phase Line Diagrams. Problems 1 through 7 involve equations of the form dy/dt = f(y). In each problem, sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. 1. dy/dt = y(y - 1)(y-2), yo≥ 0arrow_forwardPopulations of owls and mice are modeled by the equations (equations in picture). Answer the following questions. 1. Which of the variables, x or y, represents the owl population and which represents the mice population? Explain. 2. Find the equilibrium solutions and explain their significance.arrow_forwardCalculate the particular solution associated with the following dynamical equation 2 x(t) + a ¿(t) = 16t, with a=13, at t=14s.arrow_forward
- 6. For the autonomous DE: = (y - 4)y". dx a. Determine equilibrium points; b. Classify each equilibrium point as asymptotically stable, unstable, or semi-stable; c. Draw the phase line, and sketch several graphs of solutions in the xy-plane.arrow_forwarddy dx (a) Sketch the phase line for y, noting that y can be negative for this question. State the equilibrium solutions, and classify each as stable, unstable, or semi-stable. What will happen as x→ ∞ to solutions that have the following initial values: (b) (c) (i) y(0) = -2? (ii) y(0) 0.3 ? (iii) y(1) = 2 ? Consider the autonomous DE given by = = · ƒ(y) = y(y − 1)²(y + 1).arrow_forward[11] For each of the following equations, find general solutions; solve the initial value problem with initial condition x₁ (0) = -1, x₂ (0) = 2; • sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. -2 -13 dx₁/dt = x₁ + x2 x₁ = x₁4x₂ (a) x' = [ X (b) { 4 dx2/dt x₂ = 5x₁²x₂ 1 = -2x1 - x2arrow_forward
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