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1 Name: MA 226 Exam 2 Show Your Work and JUSTIFY Your Responses Problem Possible Actual Score TOTAL 100

2 1.) 20 points - Short Answer (4 each) A) Consider the predator-prey system of differential equations dr = 2R 1.2RF dt df = F + 0.9RF dt where R is the population size of rabbits and F is the population size of foxes. Suppose that, in adddition to the existing factors that affect populations, foxes migrate to an area if there are more than 3 times as many rabbits as foxes in that area, and they move away if there are fewer than 3 times as many rabbits as foxes in that area. Provide the new system that takes this into account. B) Convert the second order DE 2y + 3y + 5y = 0 to a first order system. C) Find the general solution of the second order DE y = 0.

3 D) Sketch the phase plane for the system dy dt = ( ) 2 0 Y 0 2 E) Consider ( ) dy 2 1 dt = Y 0 2 ( ) ( ) 1 1 with straight line solution Y = e 2t. Show that Y 0 = te 2t is NOT a solution to the 0 system. Don t spend too much time on this problem! BONUS: What is the Julia set of the map F (z) = z 2? What is the filled Julia set? If you don t remember, name your favorite Julia set.

4 2.) 16 points Given dy dt = ( ) 5 4 Y 9 0 a) Find all equilibrium solutions. b) Calculate the eigenvalues and classify the system. Be careful! If you find the wrong eigenvalues, you will likely score no partial credit for any subsequent work. c) Find the eigenvectors. d) Find the general solution to the system. e) Draw the phase portrait for the system. f) Find the particular solution to the initial condition Y(0) = (1, 0). Sketch the particular solution on the phase plane.

5 3.) 18 points Given ( ) dy 0 1 dt = Y 0 3 a) Find all equilibrium solutions. b) Calculate the eigenvalues and classify the system. Be careful! If you find the wrong eigenvalues, you will likely score no partial credit for any subsequent work. c) Find the eigenvectors. d) Find the general solution to the system. e) Draw the phase portrait for the system. f) Find the particular solution to the initial condition Y(0) = (1, 0). Sketch the particular solution on the phase plane.

6 4.) 16 points Given dy dt = ( ) 0 2 Y 5 2 For this system, λ is an eigenvalue with eigenvector V. ( ) 2 λ = 1 + 3i with V = 1 + 3i a) Classify the system. b) Find the general solution to the system. c) Find the natural period and frequency of any solution to this system. d) Determine the orientation of the system. e) Draw the phase portrait for the system. f) Find the particular solution to the initial condition Y(0) = (1, 0). Sketch the particular solution on the phase plane.

7 5.) 10 points Mario and Luigi are battling on a Mario Kart course, but their go-karts are completely out of gas and their steering wheels are broken. This means their go-karts can not move on their own. Fortunately, the course is covered in acceleration arrows. A given arrow imparts the same movement to a go-kart at any point in time. Furthermore, the brothers tires are covered in wet paint so they leave a trail behind them (Mario red and Luigi green, obviously). i) If Mario and Luigi start at different points on the course, will they ever collide? I could try to extend the analogy to one of them losing a balloon but I won t. ii) If Mario and Luigi start at different points on the course, but at some point Luigi drives over a red paint trail, will they ever collide? What can we say about their paths? iii) If Luigi drives over a green paint trail, what can we say about his path? iv) What concept from the course is this goofy analogy supposed to illustrate? Be specific.

8 6.) 20 points Below is a bank of options to fill in the table on the next page. The rows do NOT mean anything - for example, a center may or may not be associated with complex eigenvalues. classification eigenvalues type of harmonic oscillator center complex with the real part < 0 critically damped half spiral toward the origin purely imaginary with no real component overdamped sink real with λ 1 < λ 2 < 0 undamped spiral sink one repeat eigenvalue with λ < 0 underdamped Below are 4 x(t), y(t) graphs to be matched. The graphs from C and D are so alike that you can use those interchangeably.

9 phase portrait x, y graphs classification eigenvalues type of HM Which of these 4 systems can not actually exist as a harmonic oscillator in our universe? Why not?

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