Name Class TimeMath 2250 Extra Credit ProblemsChapter 9Fall 2010Due date: Submit these problems by the day after classes end. Records are locked on that date and only corrected,never appended. These extra credit problems can replace any missed problem for the entire semester.Submitted work. Please submit one stapled package per problem. Kindly label problems Extra Credit . Labeleach problem with its corresponding problem number, e.g., Xc9.1-4 . You may attach this printed sheet to simplifyyour work.Problem Xc9.1-4. (Phase Portraits)Find the equilibrium points for the system. Plot a phase diagram using the maple code below.dxdt= x − 2y + 3,dydt= x − y + 2.Example: Plot the phase diagram of u0=1 20 3u +45using maple.with(DEtools):equilEQ:=[0=x+2*y+4,0=3*y+5];solve(equilEQ,{x,y});# find diagram center (a,b)a:=-2/3;b:=-5/3;de:=[diff(x(t),t)=x(t)+2*y(t)+4,diff(y(t),t)=3*y(t)+5];ic:=[[x(0)=0,y(0)=-1],[x(0)=-1,y(0)=-1.5],[x(0)=0.5,y(0)=-2],[x(0)=0.5,y(0)=-1.5],[x(0)=-0.7,y(0)=-1.7]];DEplot(de,[x(t),y(t)],t=-10..10,ic,x=a-2..a+2,y=b-2..b+2,stepsize=0.05);The plot can also be done maple version 12+ with the Phase Portrait tool. Start at the TOOLS menu, then TASKS−→ BROWSE −→ DIFFERENTIAL EQUATIONS. See the problem notes at the course web site for Chapter 9.Problem Xc9.1-8. (Equilibrium Points)Find the equilibrium points for the system. Plot a phase diagram. The graph window should include the three equilibriumpoints.dxdt= x − 2y,dydt= 4x − x3.Problem Xc9.1-18. (Stability)Determine if the equilibrium point (0, 0) is stable, asymptotically stable, or unstable. Identify the equilibrium point asa node, saddle, center or spiral by examination of its computer-generated direction field.(a) x0= y, y0= −x(b) x0= y, y0= −5x − 4y(c) x0= −2x, y0= −2y(d) x0= y, y0= xProblem Xc9.2-2. (Classification by Eigenvalues)Compute the eigenvalues of A. Determine stability of equilibrium (0, 0) and classify as node (proper/improper), saddle,spiral, center.(a) A0 1−1 0(b) A1 22 1(c) A3 −24 −1(c) A1 −22 −3Problem Xc9.2-12. (Phase Portrait)Find the equilibrium point (it is unique) and plot by computer a phase diagram.dxdt= x + y − 7,dydt= 3x − y − 5.Problem Xc9.2-22. (Almost Linear System)Linearize the system at its equilibria and determine the stability and type of each. Plot a phase diagram by computerto verify the claims made.dxdt= 2x − 5y + x3,dydt= 4x − 6y + y4.Problem Xc9.3-8. (Predator-Prey System)Linearize the system at equilibrium point (0, 0). Verify that the phase diagram of the nonlinear system at (0, 0) is asaddle.dxdt= x(5 − x − y),dydt= y(−2 + x).Problem Xc9.3-9. (Predator-Prey System)Linearize the system at equilibrium point (5, 0). Verify that the phase diagram of the nonlinear system at (5, 0) is asaddle.dxdt= x(5 − x − y),dydt= y(−2 + x).Problem Xc9.3-10. (Predator-Prey System)Linearize the system at equilibrium point (2, 3). Verify that the phase diagram of the nonlinear system at (2, 3) is anasymptotically stable spiral.dxdt= x(5 − x − y),dydt= y(−2 + x).2Problem Xc9.4-4. (Almost Linear System)Linearize at (0, 0) and classify the equilibrium point (0, 0) of the nonlinear system, using a phase diagram to verify theconclusion.dxdt= 2 sin x + sin y,dydt= sin x + 2 sin y.Problem Xc9.4-8. (Almost Linear System)Linearize at all equilibria and classify the equilibrium points of the nonlinear system. Use a phase diagram to verify theconclusions.dxdt= y,dydt= sin πx − y.End of extra credit problems chapter
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