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Concept explainers
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.
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Chapter 2 Solutions
DIFFERENTIAL EQUATIONS-ACCESS
- Populations 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_forwardcharacterize the equilibrium point for the system x′= Ax and sketch the phase portrait.arrow_forwardThe equation for the moment about the origin of a one-dimensional system is M0 = 5(−3) + 2(−1) + 1(1) + 5(2) + 1(6). Is the system in equilibrium? Explainarrow_forward
- Find the equilibrium point. D(x) = (x-7), S(x) =x² O A. (7,$0) О В. (14,$49) C. $12.25 2 O D. (0,$49)arrow_forwardMatch each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) -5 2 v 1. a' = 1 5 3 ? v 2. a' = 1 -7 12 3. z' = 9 -8 ? v 4. a' -9 C D.arrow_forwardApplying variation of parameters to the DE: " – y = e* + e-*, we get the following system of equations: uj eº + u,e-² = 0 u e" – u,e-* = eª + e¬* What are u1 and u2? (U1 and uz are functions of x.) u1 = -e"; u2 %3D O u1 = ;a – ÷e-2*; u2 = = -e2 - æ 2 O uj = In sin a (e*) ; u2 = sec x (e") i+e ; uz = - fee 1 -2r 1 O uj = 4 2r 8 8arrow_forward
- Match each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) -1 ? v 1. x1 3 -8 ? v 2. x -3 1 5 ? v 3. х -5 7 A В 2 4. -3 x2/ x1arrow_forwardMatch each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) ? ✓ | 1. z ' = || a' ? 2. ': = ? 3.' = 4. a: = 11 8] -10 3 1 5 -2 1 -5 -13 10] -10 x2 A x2 с x1 (x2 B 2x2/ D Note: To solve this problem, you only need to compute eigenvalues. In fact, it is enough to just compute whether the eigenvalues are real or complex and positive or negative.arrow_forwardConsider the system of equations (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g., y=x.) (Note that there are also nullclines lying along the axes.) dx dt (b) What are the equilibrium points for the system? Equilibria = (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).) = x(2 - x - 3y) taking (x, y) > 0. dt dt dt. Recall that a nullcline of this system is a line on which = = 0. Likewise, a vertical nullcline of this system is a line on which = 0, and a dy horizontal nullcline of this system is a line on which = 0. dt (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: dy dt = y(1-2x), (c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence: If we start at the initial position (1/2,), trajectories ? ✓the point (Enter the point as an (x,y) pair, e.g., (1,2).)arrow_forward
- Discrete Mathematics and Its Applications ( 8th I...MathISBN:9781259676512Author:Kenneth H RosenPublisher:McGraw-Hill EducationMathematics for Elementary Teachers with Activiti...MathISBN:9780134392790Author:Beckmann, SybillaPublisher:PEARSON
- Thinking Mathematically (7th Edition)MathISBN:9780134683713Author:Robert F. BlitzerPublisher:PEARSONDiscrete Mathematics With ApplicationsMathISBN:9781337694193Author:EPP, Susanna S.Publisher:Cengage Learning,Pathways To Math Literacy (looseleaf)MathISBN:9781259985607Author:David Sobecki Professor, Brian A. MercerPublisher:McGraw-Hill Education
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