Slope Fields Assignment

1. The calculator-drawn slope field for the differential equation dy/dx = xy is shown in the figure below. The solution curve passing through the point (0, 1) is also shown. slope field for dy/dx = xy (a) Sketch the solution curve through the point (0, 2). (b) Sketch the solution curve through the point (0, -1). 2. The calculator-drawn slope field for the differential equation dy/dx = x + y is shown in the figure below.   slope field for dy/dx = x + y (a) Sketch the solution curve through the point (0, 2). (b) Sketch the solution curve through the point (0, -1). Draw a slope field for each of the following differential equations. Show a segment at each indicated point. Note the scales! 3. dy/dx = x 4. dy/dx = 2y slope fields for dy/dx = x and dy/dx = 2y 5. dy/dx = -y/x 6. dy/dx = x + 1 slope fields for dy/dx = y/x and dy/dx = x + 1 For problems 6-8, find the equations of the solution curves you sketched in problems 3-5. Each equation should be expressed in the form of y = f(x). Use your graphing calculator to graph each of your equations for problems 6-8 to see if those graphs match your solution curves drawn in problems 3-5.   6. dy/dx = x + 1 ∫dy = ∫(x + 1)dx y = 1/2x^2 + x + C use condition (1, 0) 0 = 1/2 + 1 + C C = -3/2 y = 1/2x^2 + x - 3/2 7. dy/dx = 2y ∫dy/y = ∫2dx ln|y| = 2x + C y = Ce^(2x) use (0, -1) -1 = Ce^0 C = -1 y = -e^(2x) 8. dy/dx = -y/x ∫dy/y = ∫-dx/x ln|y| = -ln|x| + C y = C/x use (2, -1) -1 = C/2 C = -2 y = -2/x SLOPE FIELDS At right is a slope field for the differential equation dy/dx = e^(-x) (a) Sketch the solution curve passing through the point (0, 0). (b) Find a particular solution in the form of y = f(x) to the differential equation. dy/dx = e^(-x) ∫dy = ∫e^(-x)dx y = -e^(-x) + C use (0, 0) 0 = -e^(0) + C => C = 1 y = -e^(-x) + 1 (c) Without the use of a calculator, determine whether your equation from part b represents the function which you graphed in part a. Yes, graph of y = e^x is reflected about both x-axis and y-axis and shifted up 1. 10. The slope field for a differential equation is shown at the right. Which statement is true for solutions of the differential equation? I. For x < 0 all solutions are decreasing.1 False   1. II. All solutions level off near the x-axis. True, all slopes are 0 on x-axis. III. For y > 0 all solutions are increasing. True (a) I only (b) II only (c) III only (d) II and III only (e) I, II, and III 11. The slope field for the differential equation dy/dx = (x^2y + y^3)/(4x + 2y) will have horizontal segments when (a) 2x only (b) x = -2x0 only (c) y = -x^2 only (d) y = 0 only (e) y = 0 or y = -x^2 horizontal segments when slope = 0 x^2y + y^3 = 0 y(x^2 + y) = 0 y = 0 and y = -x^2 12. Which one of the following could be the graph of the solution of the differential equation whose slope field is shown at the right? Compare each graph with slopes on slope field. (A) (B) (C) (D) (E) 13. Which statement is true about the solutions y(x) of a differential equation whose slope field is shown at the right? I. If y(0) > 0 then lim(x->∞) y(x) = ∞. False II. If -2 < y(0) < 0 then lim(x->∞) y(x) = -2. True III. If y(0) < -2 then lim(x->∞) y(x) = -2. True (A) I ONLY (B) II ONLY (C) III ONLY (D) II and III ONLY (E) I, II, and III 14. Shown at the right is the slope field for which of the following differential equations? (A) dy/dx = 1 + x (B) dy/dx = x^2 (C) dy/dx = x + y slope = 0 when x and y are the same numbers but different signs, or x + y = 0