Question

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 type of fixed point (stable node, unstable node, stable focus, unstable focus, center, or saddle point), and make a rough sketch of the solution x(t) on the axes provided on the next page. Note that your sketch only needs to be qualitatively (not quantitatively) correct.

a) [5 pts] dx dt = −x dy dt = x − 2y

b) [5 pts] dx dt = −y dy dt = x

c) [5 pts] dx dt = −y dy dt = −10x − 2y

d) [5 pts] dx dt = x dy dt = 3x − y

e) [5 pts] dx dt = x − y dy dt = 2x + y

Answer 1

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since the question was extremely long I have tried to explain each of the parts precisely. If you cannot understand any portion feel free to ask.