Math 232, Final Test, 20 March 2007 Name: Instructions. Do any five of the first six questions, and any five of the last six questions. Please do your best, and show all appropriate details in your solutions. Thank you!. Draw the phase line for the differential equation dy ( dt = y3 y ) ( y ) 2. 50 00 Identify the equilibrium points as sources, sinks or nodes. Finally, sketch solutions with the following initial conditions: (i) y(0) = 25, (ii) y(0) = 25. (iii) y(0) = 00, (iv) y(0) = 25. 2. A 400 gallon tank initially contains 00 gallons of water containing mg of dioxin per gallon. Water containing 5mg per gallon of dioxin is pumped into the tank at a rate of 4 gallons per minute, while the well-mixed water is discharged at a rate of 2 gallons per minute. How much dioxin is in the tank when the tank is full?
3. (a) Consider the one parameter family of differential equations dp dt = P 2 +P 5K that 000 represents a population of fish in a pond. Determine the bifurcation values for K, and draw representative phase lines at, and on each side of the bifurcation value(s). (b) If K = 32, describe the long term behavior of the population. In particular, determine, if possible, how many fish must be present initially for the population to survive? In the event enough fish are present for the population to survive describe the long term behavior of the population. dx dy ( 4. Consider the predator-prey system: = x + 0xy and dt dt = y y ) xy. 500 (a) Which variable represents the prey? Which variable represents the predator? Explain. (b) What will happen to the prey population long-term if there are no predators? (c) Suppose the predators found a second source of food of limited supply, say one that would give them a carrying capacity of 00. How would you modify the differential equations?
5. Find the solution to the partially decoupled system dx dt through the point (0, ). = x + and dy dt = 2xy that passes 6.(a) Find the general solution to the system dx dy = 2x + y and = x + 4y. dt dt (b) Find the specific solution to the system in (a) that passes through the point (, 2).
7. Sketch the phase portraits for the following systems of differential equations with the help of the given information about their eigenvalues and/or eigenvectors. Directions are important! (a) dy [ dt = 3 5 Y. Eigenvalues: λ = ± i. 3 (b) dy dt = [ 2 2 3 [ Y. Eigenvalues and eigenvectors: λ = 4, [ 2 ; λ =, (c) dy dt = [ 2 3 3 2 [ Y. Eigenvalues and eigenvectors: 5, and, [.
8. (a) Solve the initial value problem y 6y + 5y = 3e 2t subject to y(0) = 0 and y (0) = 0. 9. Find the general solution to y + 4y + 3y = 3 cos 3t
0. Consider the system dx dt = 00x x2 2xy and dy dt = 50y xy 6y2. (a) Find all equilibrium points. (b) Use the Jacobian to classify the equilibrium point that does not lie on the x-axis or the y-axis (as a sink, source, saddle, etc). (c) In general, when can linearization fail to accurately predict the behavior of the nonlinear system near an equilibrium point?. Find and sketch nullclines for system in question 0, and then sketch phase portrait in the first quadrant. Given that x(t) and y(t) represent populations of two species, which of the following is the most likely long term outcome? (i) both species will survive and approach stable populations, (ii) one species will go extinct, or (iii) both species will go extinct. Explain.
2. (a) Determine whether the system dx dt = x sin y + 3y2 and dy = cos y + 2x + is a dt Hamiltonian system. (b) If the system in (a) is Hamiltonian, find its Hamiltonian function H(x, y). (c) Describe briefly how a Hamiltonian function can be used to find the phase portrait of a Hamiltonian system.