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Do not write in the boxes immediately below. problem 1 2 3 4 5 6 7 8 9 total points Math U343 Final Exam Instructor’s name Your name Answers from your calculator, without supporting work, are worth zero points. The last page of this exam is a table of Laplace transforms, which you may rip off. The total number of points on the exam is 100. 1) (10 points) Find the general, explicit solution to y 2 y dy = +1 + . dx x x 2 2) (10 points) In the study of propagating waves, the variation of the wave is described by u = u(t) at time t along the wave front and satisfies du + m(t)u2 + r(t)u = 0. dt Suppose that m(t) = −3t, r(t) = −t−1 , and u(1) = 1/4. Find the explicit solution for u. 3 du + m(t)u2 + r(t)u = 0. dt This time, however, suppose that m(t) is the constant −5, and that r(t) is the constant 10. 3) As in Problem 2, consider once again the equation for the variation of a wave: a) (4 points) What are the equilibrium solutions for the resulting autonomous equation? b) (4 points) Using the same differential equation as in part a), if u(0) = 0.5, give a qualitative description of what happens to u = u(t) as time goes on. You must give a complete answer for full credit, and you must show some supporting work, but you can answer this question easily WITHOUT actually solving the differential equation. 4) (10 points) Find the general, explicit solution to x00 + 4x0 + 5x = 0. Primes denote derivatives with respect to t. 4 5) (12 points) Consider the initial value problem x00 + 5x0 + 4x = sin(t); x(0) = 0, x0 (0) = 0, where x is a function of t. a) (3 points) Describe a physical situation (i.e., write a word problem) which would give one the initial value problem above. For instance, you might refer to a mass on a spring or an electrical circuit. b) (9 points) Solve the initial value problem. 5 6) A mass of 1 kg is attached to a spring with a spring constant of 4 N/m and is also attached to a dashpot which provides 5 Newtons of resistance for each m/s of velocity. There is a constant external force of 12 Newtons acting on the mass. In addition to this constant external force, the mass is struck at time 2 seconds by an instantaneous infinite force, which causes an instantaneous change of momentum of 9 kg · m/s. This gives one the differential equation x00 + 5x0 + 4x = 12 + 9δ(t − 2). The mass starts at rest at the equilibrium position. a) (12 points) Use Laplace transforms to find the position of the mass as a function of time t, measured in seconds. IF YOU DO NOT USE LAPLACE TRANSFORMS, YOU WILL RECEIVE NO CREDIT ON THIS PROBLEM. b) (2 points) What is the position of the mass at time t = 3 seconds? 6 5 −15 2 0 7) (6 points each) Let A = −2 6 −1 −1 . Then, the reduced row echelon form of A is 3 −9 1 −1 1 −3 rref(A) = 0 0 0 0 0 −2 1 5 . 0 0 Use this information to answer the two questions below. a) Write (0, −1, −1) as a linear combination of (5, −2, 3), (−15, 6, −9), and (2, −1, 1). b) Find all of the solutions to the following system of equations. Write your answer in vector form. Remember to use the information given at the top of the page (or you will waste a lot of time). 5x1 − 15x2 + 2x3 = 0. −2x1 + 6x2 − x3 = −1. 3x1 − 9x2 + x3 = −1. 7 1 0 0 8) (12 points) Let A = −4 7 2 . Find the eigenvalues of A and, for each eigenvalue, find a basis for the 10 −15 −4 corresponding eigenspace. 8 9) Consider the following linear system of differential equations. x1 , x2 , and x3 are functions of t. x01 = x1 . x02 = −4x1 + 7x2 + 2x3 . x03 = 10x1 − 15x2 − 4x3 . a) (2 points) If this system of equations is written in the matrix form d~x = A~x, what is the matrix A? dt b) (10 points) Use your answer to Problem 8 to find the general solution to the system of differential equations. 9 function 1 t tn ta eat tn eat cos(kt) sin(kt) cosh(kt) sinh(kt) eat cos(kt) eat sin(kt) u(t − a) δ(t − a) transform 1 s 1 s2 n! sn+1 Γ(a + 1) sa+1 1 s−a n! (s − a)n+1 s s2 + k 2 k s2 + k 2 s 2 s − k2 k s2 − k 2 s−a (s − a)2 + k 2 k (s − a)2 + k 2 e−as s −as e