MATH 251ExaminationJuly 16, 2009Name:Student Number:Section:This exam has 9 questions for a total of 100 points. In order to obtain full credit for partialcredit problems, all work must be shown. Credits will not be given for an answernot supported by work. The point value for each question is in parentheses to the right of thequestion number.You may not use a calculator on this exam. Please turn off and put away yourcell phone.The last sheet of the booklet can be removed. Be careful to remove only the last page ofthe examination.1:2:3:4:5:6:7:8:9:Total:Do not write in this box.MATH 251 EXAMINATION July 16, 20091. (10 points) For parts (a) through (e) below, a list of differential equations is given. For eachpart, write down the letter corresponding to the equation on the list with the specified proper-ties. There is only one correct answer to each part.A. y0= 2y + tB. y0= e2y−tC. y0= ey− 1D. y00+ 4y0− 5y = 2E. y00+ ety0+ t2y = 0F. y00− 4y =tyG. y000+ 3y00+ 3y0+ y = t5+ ln tH. y000+ y0y = e−2tsin 5t(a) Second order homogeneous linear equation.(b) Third order nonlinear equation.(c) Second order nonhomogeneous linear equation.(d) First order linear equation.Page 2 of 10MATH 251 EXAMINATION July 16, 20092. (10 points) Solve explicitly for y(t) in the following initial value problemet− y y0= 0; y(0) = 1.Page 3 of 10MATH 251 EXAMINATION July 16, 20093. (10 points) A tank is filled with 200 liters of a solution containing 100 grams of salt. A solutioncontaining a concentration of 2 g/liter salt enters the tank at the rate 4 liters/minute and thewell-stirred mixture leave s the tank at the same rate. Set up the initial value problem for theamount of salt in the tank at time t, find the particular solution and find the limiting amountof salt in the tank as t → ∞.Page 4 of 10MATH 251 EXAMINATION July 16, 20094. (15 points) For the following initial value problem ty0= 3y + t; y(4) = −1(a) Without solving it, find the maximum interal on which we are guaranteed that the problemhas a uniqe solution.(b) Solve the initial value problem.Page 5 of 10MATH 251 EXAMINATION July 16, 20095. (15 points) Find the particular solution to y00− 5y0+ 4y = 0, y(0) = 2, y0(0) = −1. What isthe behavior of the solutions when t → +∞?Page 6 of 10MATH 251 EXAMINATION July 16, 20096. (5 points) Show y1(t) = t2and y2(t) = t3, t > 0 are linearly independent by calculating theWronskian.7. (10 points) Provided y1(t) = t solves the equation t2y00+ 2ty0− 2y = 0, t > 0, write down thegeneral solution of the above equation.Page 7 of 10MATH 251 EXAMINATION July 16, 20098. (10 points) What is the form of the general solution to the following equation?Do not solve for the constants!y00− y = cos(2t) + 3tet− 4 sin(t)Page 8 of 10MATH 251 EXAMINATION July 16, 20099. (15 points) (a) Circle the correct answer. By definition {f (t)} =(i)R∞0estf(t)dt(ii)R∞0e−stf(t − c)dt(iii)R∞0e−stf(t)dt(b) Solve the following equation using Laplace’s transform.y0− y = et, y(0) = 1.No credits will be given for other methods.Page 9 of 10MATH 251 EXAMINATION July 16, 2009Table of Laplace Transformf(t) F (s) commentsc1ss > 0tnn!sn+1eat1s − as > asin ωtωs2+ ω2s > 0cos ωtss2+ ω2s > 0tneatn!(s − a)n+1n = positive integer, s > aPage 10 of
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