Week7: Problem 1 Previous Problem Problem List Next Problem (1 point) Match the solutions to the differential equations. If there is more than one solution to an equation, select the answer that includes all solutions. 1. d'y = -36y dx2 dy 2. = 6y dx 3. d'y = 36y dx2 ay 4- dx = -6y A. y = eox B. y = e-ox or y = eox C. y = e-6x D. y = sin(6x) E. y = 6 sin(x) F. y = sin(6x) or y = 6 sin(x)Week7: Problem 2 Previous Problem Problem List Next Problem (1 point) Let A and k be positive constants. Which of the given functions is a solution to - = k(y - A)? OA. y = A + Ce-kt OB. y = A + Cekt OC. y = A- + Ce-Akt OD. y = -A + Ce-kt OE. y = -A + Cekt OF. y = A-1 + CeAktWeek7: Problem 3 (1 point) Find a positive value of k for which y = cos(kt) satisfies dzy + 9 = 0. d:2 y \"VGGRh ProBIem 2! (1 point) Consider the two slope fields shown, in figures 1 and 2 below. \\\\\\t//// \\\\\\\\--222/ \\ '\\ '\\ '1 i J I j . I I f l 1 / figure 2 On a print-out of these slope fields, sketch for each three solution curves to the differential equations that generated them. Then complete the following statements: For the slope field in figure 1, a solution passing through the point (4,-1) has a 7 v slope. For the slope field in figure 1, a solution passing through the point (2,0) has a ? v slope. For the slope field in figure 2, a solution passing through the point (1,0) has a ? v slope. For the slope field in figure 2, a solution passing through the point (0,-1) has a ? v slope. Week7: Problem 5 Previous Problem Problem List Next Problem (1 point) The slope field for the equation y = -x - y is shown below On a print out of this slope field, sketch the solutions that pass through the points (i) (0,0); (ii) (3, 1); and (iii) (1,0). From your sketch, what is the equation of the solution to the differential equation that passes through (1,0)? (Verify that your solution is correct by substituting it into the differential equation.) y =(1 point) The slope field for the equation d P/dt = 0.05 P(20 - P), for P 2 0, is shown below. On a print out of this slope field, sketch the solutions that pass through (0, 0); (2, 8); (8, 2); (-9.5, 2); (-4, 24); and (-4, 20). For which positive values of P are the solutions increasing? Increasing for: (Give your answer as an interval or list of intervals, e.g., if P is increasing between 1 and 5 and between 7 and infinity, enter (1,5), (7, Inf).) For what positive values of P are the solutions decreasing? Decreasing for: (Again, give your answer as an interval or list of intervals, e.g., if P is decreasing between 1 and 5 and between 7 and infinity, enter (1,5), (7, Inf).) What is the equation of the solution to this differential equation that passes through (0,0)? P = If the solution passes through a value of P > 0, what is the limiting value of P as t gets large? P -Week7: Problem 7 Previous Problem Problem List Next Problem (1 point) The figure below shows the slope field for the equation y' = sin(x) cos(y). On a print out of this slope field, sketch the solutions that pass through the points (i) (0, -2) (ii) (0, It/2) What is the equation of the solution passing through (0, (2n + 1)x/2), where n is any integer? y =Week7: Problem 8 (1 point) Consider the differential equation y' = x y. Use Euler's method with Ax = 0.1 to estimate y when x = 1.4 for the solution curve satisfying y(1) = 1 : Euler's approximation gives y(1.4) z Use Euler's method with Ax = 0.1 to estimate y when x = 2.4 for the solution curve satisfying y(1) = 0 : Euler's approximation gives y(2.4) :5 Week7: Problem 9 (1 point) Consider the solution of the differential equation y' = y passing through y(0) = 1.5. A. Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,1.5). B. Use Euler's method with step size Ax = 0.2 to estimate the solution at x = 0.2, 0.4, , 1, using these to fill in the following table. (Be sure not to round your answers at each step!) C. Plot your estimated solution on your slope field. Compare the solution and the slope field. Is the estimated solution an over or under estimate for the actual solution? 0 A. over 0 B. under D. Check that y = 1.5e'1" is a solution to y' = y with y(0) = 1.5. Week7: Problem 10 (1 point) Consider the differential equation with initial condition y(0) = 1. A. Use Euler's method with two steps to estimate y when x = 1: y(1) =5 (Be sure not to round your calculations at each step!) Now use four steps: y(1) % (Be sure not to round your calculations at each step!) B. What is the solution to this differential equation (with the given initial condition)? y = C. What is the magnitude of the error in the two Euler approximations you found? Magnitude of error in Euler with 2 steps = Magnitude of error in Euler with 4 steps = D. By what factor should the error in these approximations change (that is, the error with two steps should be what number times the error with four)? factor = (How close to this is the result you obtained above?)