4. Explicit Limit Cycles. Find, using plane-polar coordinates or otherwise, the explicit limit cy...
Problem 3. Consider the following continuous differential equation dx dt = αx − 2xy dy dt = 3xy − y 3a (5 pts): Find the steady states of the system. 3b (15 pts): Linearize the model about each of the fixed points and determine the type of stability. 3b (15 pts): Draw the phase portrait for this system, including nullclines, flow trajectories, and all fixed points. Problem 2 (25 pts): Two-dimensional linear ODEs For the following linear systems, identify the...
HELP Math 244 Fake Final Exam 2e syatem 13. 116 points Consider the non-linear system dt a) Use r -x2 + y2 to write dr as a function of r. b) This system has a limit cycle. Identify it, and determine whether it is stable or unstable. dr (t may help to draw the phese ine for the DE di )) e) The systemhas only one critical point, at the origin. (You do not need to prove this.) What, specifically,...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the system (b) Classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). (c) Plot the phase portrait of the system containing a trajectory with direction as t-oo whose initial value is X(0) (0,6)7 and any other trajectory with direc- tion. (You do not need to draw solution curves explicitly.) Consider the plane...
3. [4 marks] Using cylindrical polar coordinates, or otherwise, find the value of the surface integral 1 = []6.2? ds, where S is the part of the cone z = z? + y that lies between the planes z = 2 and 2=5.
(1 point) Consider the system of differential equations dx dt = -1.6x + 0.5y, dy dt = 2.5x – 3.6y. For this system, the smaller eigenvalue is -41/10 and the larger eigenvalue is -11/10 [Note-- you may want to view a phase plane plot (right click to open in a new window).] If y' Ay is a differential equation, how would the solution curves behave? All of the solutions curves would converge towards 0. (Stable node) All of the solution...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...