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For the system
(a)show that the system is Hamiltonian with Hamiltonian function
(b) sketch the level sets of H, and
(c) sketch the phase portrait for the system. Include a description of all equilibrium points and any saddle connections.
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Chapter 5 Solutions
Differential Equations
- Obtain the phase plane of the system x´= 1 - x2y´ = xy Solve for y=y(x). Please be as clear and legible as possible. Show all the steps. Thank you.arrow_forwardPlease solve for linear system (d)arrow_forwardusing the picture below, answer the question, but focus on a more detailed explanation of your thought-process. In other words, do not just post an image of your work, but take the time to rewrite it and make it as easy to follow as possible. (use explanations/ complete sentences on how you got the answer and what process were used)arrow_forward
- Please hand write your work if possible or make it clear. I have trouble reading typed answers. Thank you.arrow_forwardB pleasearrow_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_forward
- Solve the systemarrow_forwardFind the point (x,y) on the line y=x that is equidistant from the points (3,8) and (−10,−10)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
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage