Name: MA 226 FINAL EXAM Show Your Work Problem Possible Actual Score 1 36 2 8 3 8 4 8 5 8 6 8 7 8 8 8 9 8 TOTAL 100
1.) 30 points (3 each) Short Answer: The answers to these questions need only consist of one or two sentences. Partial credit will be awarded only in exceptional situations. I) Solve the initial-value problem: dy dt 3y = 0, y(0) = 5. II) A series circuit has a capacitor of 0.25x10 3 farad, a resistor of 400 ohms, and an inductor of 0.125 henry. The initial charge on the capacitor is 5x10 6 coulomb and there is no initial current. At time t = 2 seconds, a switch is flipped and a constant voltage of 1 V is added to the system. Write down a second order differential equation and forcing function that models this circuit. Recall: V = IR, Q = V C, V = L di dt, I = dq dt. III) Locate the bifurcation value(s) of α for dy y6 2y 3 + α
1. continued) Short Answer: The answers to these questions need only consist of one or two sentences. Partial credit will be awarded only in exceptional situations. IV) Solve for L[y] given 2 d2 y dt 2 + 11dy dt + 13y = 7u 5(t), y(0) = 1, y (0) = 3. V) If the nonhomogenous portion of a second-order linear DE is: e 2t cos 5t, and the required guess for the particular solution is Re[te ( 2+5i)t ], what must the left hand side of the DE be? VI) Given that a system has one eigenvalue of λ = 4 and the other eigenvalue is unknown, list the possible behaviors of the origin.
1. continued) Short Answer: The answers to these questions need only consist of one or two sentences. Partial credit will be awarded only in exceptional situations. VII) Sketch the direction field for the system: y 2 where Y = x(t) 1 + x + y y(t). VIII) Below is the result of Euler s Method, from 0 t 5, where t = 0.5, for the IVP: dy dt = (3 y)(y + 1), y(0) = 0. What is wrong with the estimates that Euler s Method gives?
1. continued) Short Answer: The answers to these questions need only consist of one or two sentences. Partial credit will be awarded only in exceptional situations. IX) For the differential equation: dy y 1 + t + t2 1 Calculate µ(t). Why do we multiply both sides of the DE by this factor? (i.e. How does it help us solve the DE?) X) Consider a vat with a capacity of 100 gallons of water. The vat currently holds 20 gallons of pure water containing 3 parts per million of substance X. The solution is well mixed. Each minute, 6 gallons of water containing 0.5 ppm of X is added to the vat. At the same time, well-mixed solution exits the vat at a rate of 2 gal/min. Write - DO NOT SOLVE - an initial-value problem modeling the situation. Do not forget to give a time interval if necessary.
1. continued) Short Answer: The answers to these questions need only consist of one or two sentences. Partial credit will be awarded only in exceptional situations. XI) Find all equilibrium points of the system: dx dt = y x and dy x x3 + y. XII) Below is the phase portrait for the system from the critically damped spring-mass system: d 2 y dt 2 + 6dy dt + 9y. Does the particular solution corresponding to an initial position of y = 1 and initial velocity of v = 2 cross the rest position 0, 1, or many times?
2.) 8 points Solve the given linear system. For the sake of partial credit, that means: i) compute the eigenvalues and classify the origin; ii) for each eigenvalue, compute the associated eigenvectors (FYI, 3 1.75); iii) compute the general solution; and iv) sketch the phase portrait. 1 1 Y 2 1
3.) 8 points Solve the given linear system. For the sake of partial credit, that means: i) compute the eigenvalues and classify the origin; ii) for each eigenvalue, compute the associated eigenvectors; iii) compute the general solution; and iv) sketch the phase portrait. 3 1 Y 1 1
4.) 8 points Solve the given linear system. For the sake of partial credit, that means: i) compute the eigenvalues and classify the origin; ii) for each eigenvalue, compute the associated eigenvectors; iii) compute the general solution; and iv) sketch the phase portrait. 2 1 Y 0 1
5.) 8 points Solve the given linear system. For the sake of partial credit, that means: i) compute the eigenvalues and classify the origin; ii) for each eigenvalue, compute the associated eigenvectors; iii) compute the general solution; and iv) sketch the phase portrait. 0 2 Y 0 1
6.) 6 points Consider the linear system AY, where A is a 2 x 2 matrix with real entries such that it has 1 eigenvalue λ = 2 + 5i with associated eigenvector v =. 4 3i i) compute the general solution. ii) sketch the phase portrait. Note that if λ = 2 + 5i is an eigenvalue, then the homogenous DE must be d2 y dt 2 + 4dy dt + 9y. iii) compute the particular solution for Y 0 = 2. 1
7.) 8 points Consider the one-parameter family of linear systems depending on a. 1 a 2 a 1 0 Y i) sketch the corresonding curve in the trace-determinant plane; ii) identify the values of a where the type of system changes (i.e. the bifuracation values). When calculating intersections, it might be helpful to note that 320 = 8 5 and 8 5 > 16. iii) discuss different types of behaviors exhibited by the system as a takes different values.
8.) 8 points Consider the following species model: 150x(1 x ) 3xy 150 100y(1 y ) 2xy 100 i) Compute all the equilibrium points. ii) For each eq. pt. that is not the origin, linearize each equilibrium point and classify the behavior of each eq. pt. locally. iii) What kind of population model system is this? Do we expect to see both species survive in the long run?
9.) 6 points Consider the following species model again: 150x(1 x ) 3xy 150 100y(1 y ) 2xy 100 i) Find the x-nullclines and y-nullclines; ii) Sketch the first quadrant with the nullclines; iii) Determine the direction of solutions on each nullcline; and