Name Student ID Final MAT 22B, Friday August 4, 2017 Instructions: This exam is closed notes, closed books, no calculators and no electronic devices of any kind. There are five problems worth 35 points each and one extra credit worth 15 points. If a problem has multiple parts, it may be possible to solve a later part without solving the previous parts. Solutions should be written neatly and in a logically organized manner. Please show the steps you took to arrive at the solution. Partial credit will be given if the student demonstrates an understanding of the problem and presents some steps leading to the solution. Correct answers with no work will be given no credit. The back sheets may be used as scratch paper but will not be graded for credit. 1 2 3 4 5 Total E 1 Problem 1. Consider the differential equation t2 y 00 2ty 0 + 2y = 0, 0 < t < 1. a. (5 points) Show that y1 (t) = t is a solution. b. (10 points) Use Abel's formula to show that the Wronskian of any two solutions of the given equation is W (t) = y1 (t)y20 (t) y10 (t)y2 (t) = c1 t2 . c. (10 points) Use the results of parts a. and b. to show that a second solution is y2 (t) = t2 . d. (10 points) Can y1 (t) = t and y2 (t) = t2 be a fundamental set of solutions of an equation y 00 + p(t)y 0 + q(t)y = 0, 1 < t < 1, where p and q are continuous functions on the interval 1 < t < 1? 2 Problem 2. a. (15 points) Find the general solution to the system of equations: dx1 = x1 + x2 dt dx2 = 4x1 2x2 . dt b. (10 points) Sketch the phase portrait and trajectories of the system in the (x1 , x2 ) plane. Indicate the trajectory that goes through the point (5, 0). Classify the equilibrium point at the origin (0, 0). c. (10 points) Find the solution that goes through the point (5, 0) at t = 0. 3 Problem 3. a. (10 points) Let y1 (t) = et cos(3t) and find a second order, linear, constant coefficient, homogeneous differential equation such that y1 is a solution. b. (15 points) Find the general solution of the equation from a. c. (10 points) Describe a system in the real world that can be modelled using the equation from a. Define the variables in the model and discuss the significance of each term in the equation. 4 Problem 4. Recall that the Laplace transform of f (t) is given by Z L[f ](s) := ets f (t)dt. 0 1. (10 points) Compute L[et ](s) and L[t](s). For which values of s are your computations valid? 2 2. (10 points) Find a formula for L[ dtd f ](s) and L[ dtd 2 f ](s). 3. (15 points) Use the Laplace transform the solve the initial value problem y 00 y = et , y(0) = 0, 5 y 0 (0) = 0. Problem 5. a. (15 points) Find the general solution to the equation y0 + 1 y = . x x Using the theorem of existence and uniqueness, state how the domain of the general solution is dependent on the choice of initial condition. b. (10 points) In the limit as x the general solution approaches a particular solution (x). Find the solution (x) and compute (0). c. (10 points) Draw a direction field with rate function f (x, y) = 1y for x 0. Sketch intex gral curves for the graph of the solutions with initial condition y(1) = 1, y(1) = 1 and the function (x) you found in part b. Hint: Start with isolines for c = 2, 1, .5, 0, .5, 1, 2 and add more as needed. 6 Extra Credit. Let A be a self-adjoint n-by-n matrix. Recall that for a self-adjoint matrix there exists an orthonormal basis of eigenvectors x(1) , x(2) , . . . , and x(n) , with corresponding eigenvalues 1 , 2 , . . . , and n that are real. The order is chosen so that 1 is the largest eigenvalue, 1 k for k = 2, 3, . . . , n. a. (10 points) Show that for all vectors x (x, Ax) 1 . (x, x) P (k) Hint: Write x = nk=1 ck x(k) where Pnck = (x , x). Use this formula to compute Ax, and also use it to show that (x, x) = k=1 ck ck . b. (5 points) What is the general solution to the system of equations x0 = Ax? If 1 < 0 determine the behavior of the solution as t . 7