A slope field for the differential equation is shown on a square grid that includes…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A slope field for the differential equation dxdy=x−y is shown on a square grid that includes all lattice points with integer coordinates from −1 to 1 in both the x- and y-directions.
At each lattice point a short line segment with slope equal to x−y has been drawn. A solution curve passes through the origin.
(a) Compute the slopes of the segments drawn at the points (0,0), (21,0), and (1,0). Explain how the signs of these slopes indicate the initial direction of the solution curve beginning at the origin.
(b) Without making any numerical calculations beyond those in part (a), determine whether the y-coordinate of the solution curve at x=1 is likely to be positive, negative, or zero. Provide a qualitative justification that refers explicitly to the pattern of slopes in the field.
(c) Describe a step-by-step graphical procedure, using a constant horizontal increment of Δx=0.25, that could be followed on the slope-field grid to obtain an approximate value of y(1) for the solution passing through the origin. Your answer should identify what is done at each step and why the procedure is expected to produce a reasonable estimate.