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.
18.
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Differential Equations
- 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_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_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_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. 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_forwardОк Denote the owl and wood rat populations at time k by x = Rk where k is in months, Ok is the number of owls, and R is the number of rats (in thousands). Suppose OK and RK satisfy the equations below. Determine the evolution of the dynamical system. (Give a formula for XK.) As time passes, what happens to the sizes of the owl and wood rat populations? The system tends toward what is sometimes called an unstable equilibrium. What might happen to the system if some aspect of the model (such as birth rates or the predation rate) were to change slightly? Ok+1 = (0.2)0k + (0.5)RK Rk+1=(-0.16)0k + (1.1)Rk Give a formula for xk- xk = 4 ( D +C₂1arrow_forwardT16. Produce the general solution of the dynamical system .90 .01 .09 Xk+1 Axk when A= .01 .90 .01 .09 .90 .09arrow_forward
- Applying variation of parameters to the DE: " – y = et + e-*, we get the following system of equations: uj e + uze- = 0 uj e* – uze- = e* + e¬* What are uj and u2? (u1 and uz are functions of x.) 2x O u - u1 = -e*; u2 e +e-2; uz = - te 2a: = In. 8 8. = In O In sin a (e"); u2 = sec a (e-*)arrow_forwardX: X(0) = -12 Draw the phase plane portrait, component graphs and state the equilibrium. (). Solve the system : X =/10 -5 2arrow_forwardDetermine all critical points of the system of equations- O (0,0) and (-5,6) O(0,0), (0, 1), and (-5,6) ° (0.0), (0.7). and (-5,6) O (1,0) and (-5,6) ° (0.0), (0.2). dx (1, 0), and (-5,6) dt = x - x²- -xy, and dy dt = 7y - xy - 2y².arrow_forward
- How does the phase portrait behave for each of the equilibrium points after we linearize the nonlinear system?arrow_forwardSolve and identify the type of equilibrium point at the origin. x' 3 5.arrow_forwardIn Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let's modify those equations as follows: dR = 0.1R(1 – 0.0001R) – 0.003RW dt dW -0.01W + 0.00004RW dt Find all of the equilibrium solutions. Enter your answer as a list of ordered pairs (R, W), where Ris the number of rabbits and W the number of wolves. For example, if you found three equilibrium solutions, one with 100 rabbits and 10 wolves, one with 200 rabbits and 20 wolves, and one with 300 rabbits and 30 wolves, you would enter (100, 10), (200, 20), (300, 30). Do not round fractional answers to the nearest integer. Answer =|arrow_forward
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