de la cruz (cld2889) - HW03 - berg - (54070) This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page - find all choices before answering. 001 003 10.0 points Find all values of r for which the function y = ert satisfies the differential equation y + 12y + 35y = 0. 10.0 points Find all values of k that don't result in a zero function for which the function y = sin kt satisfies the differential equation y + 25y = 0 1. k = 25, 25 1 1. r = 12, 35 2. r = 7 3. r = 7, 5 4. r = 5, 7 2. k = 5, 5 5. r = 35, 12 3. k = 5 6. r = 12 4. k = 25 004 10.0 points Which of the following families of functions is the solution to the differential equation 5. k = 5 6. k = 25 y = 4xy ? 002 10.0 points Find all nonzero values of k for which the function y = A sin kt + B cos kt satisfies the differential equation 1. y = x4 + C 2. y = Ce4x 2 y + 16y = 0 for all values of A and B. 3. y = Ce2x 1 1. k = 4 4. y = Cx 4 2. k = 16 5. y = Ce2x 3. k = 4, 4 4. k = 16 5. k = 16, 16 6. k = 4 2 005 10.0 points The family of solutions to the differential 2 equation y = 6xy is y = Ce3x . Find the solution that satisfies the initial condition y (3) = 5. 1. y = 5e3(x3) 2 de la cruz (cld2889) - HW03 - berg - (54070) 2. y = 5e3(x 3. y = e3(x 2 2 9) 4. y = 5e3(x 2 +9) 6 5. y = 5e 006 10.0 points The family of solutions to the differential 1 . equation y = 3y 2 is y = 3x+C Find the solution that satisfies the initial condition y(5) = 3. 2. y(x) = 008 10.0 points Which of the following families of functions is the solution to the differential equation y = 3xy 3 ? 9) 3(x2 9) 1. y(x) = 2 1 9(x + 5) + 3 3 3(x + 5) + 1 1. y = 2. y = 1 C + 3x2 3. y = 4. y = 5. y = 1 C 3x2 3 C 3x2 1 C + 3x2 1 C 3x2 3. y(x) = 1 3(x + 5) + 3 009 10.0 points Which of the following answers lists all constant solutions to the equation 4. y(x) = 9 3(x + 5) + 3 dy = y 4 8y 3 + 15y 2 ? dt 5. y(x) = 3 9(x + 5) + 1 007 10.0 points The family of solutions to the differential 1 . equation y = xy 3 is y = C x2 Find the solution that satisfies the initial condition y(4) = 3. 1 1. y = p 1 9(x2 16) 3 2. y = p 1 9(x2 16) 1 3. y = p (x2 16) + 1 3 4. y = p 1 (x2 16) 3 5. y = p 1 3(x2 16) 1. y = 8, 15 2. y = 8, 0, 15 3. y = 3, 5 4. y = 0, 3, 5 5. y = 0 010 10.0 points A function y(t) satisfies the differential dy equation = y 4 15y 3 + 56y 2 . dt For what values of y is y strictly increasing? 1. (, 0) (8, ) 2. (, 0) (0, 7) (8, ) de la cruz (cld2889) - HW03 - berg - (54070) 3 Using the above graph of y(x), choose the equation whose solution satisfies the initial condition y(0) = 2. 3. (7, 8) 4. (, 0) (7, 8) 1. y = y x 5. (8, ) 2. y = y 3 x3 011 10.0 points The table gives estimates of the world population, in millions, from 1750 to 2000. 3. y = y 2 x2 4. y = y 1 Year Population Year Population 1750 680 1900 1590 1800 860 1950 2580 1850 1200 2000 5890 5. y = y x2 013 10.0 points Consider the graph Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900. 6 1. P = 1367 million 4 2. P = 1389 million 2 x 3. P = 1334 million 4 4. P = 1376 million 2 2 2 5. P = 1301 million Using the above graph of y(x), choose the equation whose solution is graphed and satisfies the initial condition y(0) = 3. 012 10.0 points Consider the graph 6 1. y = y 2 x2 y 2. y = y x 1 4 3. y = y x 2 x 4 y 2 2 2 4. y = y 1 5. y = y 2 x 6. y = y + 3.5 de la cruz (cld2889) - HW03 - berg - (54070) 4 4. 3 7. y = y x 2 y 2 014 10.0 points 2 Use the direction field of the differential equation y = y 3 to sketch a solution curve that passes through the point (0, 1). y 2 2 x 2 x 2 2 1. 2 x 5. y 2 2 y 2 2 2. 2 2 x 2 2 y 2 y 2 6. 3. 2 2 2 2 x x 2 2 015 10.0 points Consider the following graph of y(x) de la cruz (cld2889) - HW03 - berg - (54070) 5 3. 3 y 6 4. 3 5. 4 6. Doesn't exist. 2 x 2 2 Choose the equation whose solution is graphed and satisfies the initial condition y(0) = 2. 017 10.0 points Use Euler's method with step size 0.1 to compute the approximate y-values y1 , y2 , and y3 of the solution of the initial-value problem y = 1 + 6x 3y, y(1) = 2. 1. y1 = 2.100, y2 = 2.224, y3 = 2.062 2. y1 = 2.100, y2 = 1.950, y3 = 2.456 1. y = x sin(2y) 3. y1 = 2.100, y2 = 2.470, y3 = 2.314 2. y = y sin(2x) 4. y1 = 2.100, y2 = 2.230, y3 = 2.381 3. y = y sin(4x) 5. y1 = 2.100, y2 = 2.580, y3 = 2.305 4. y = y cos(2x) 018 10.0 points Use Euler's method with step size 0.4 to estimate y(1.2), where y(x) is the solution of dy + x2 y = 7x2 , the initial-value problem dx y(0) = 2. 5. y = y sin(x) 6. y = y sin(2x) 016 10.0 points A solution y(t) of the differential equation y = y 3 9y satisfies the initial condition y(0) = 1.5. Find lim y(t). t 1. 0 2. 1. y(1.2) 2.289 2. y(1.2) 8.957 3. y(1.2) 4.013 4. y(1.2) 3.518 5. y(1.2) 8.57