MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Consider the differential equation dxdy=x−y. The axes shown on the accompanying coordinate grid are scaled so that one unit in the x– and y–directions corresponds to one square. Slopes have been omitted at all integer lattice points in the square region −2≤x≤2, −2≤y≤2.
Answer the following parts on your own paper.
(a) At every integer lattice point in the given region, draw a small line segment whose slope equals the value of dxdy at that point. Label the axes clearly.
(b) On the same set of axes, sketch the particular solution that satisfies the initial condition y(1)=−1. Draw the solution curve so that it extends across the entire square region and aligns with the line segments you sketched in part (a).
(c) Using your sketch, determine the intervals of x for which the solution is increasing and concave down. Justify each conclusion with reference to the geometric features of the slope field or analytic properties of the differential equation.
(d) Using only the behavior visible in your sketch and the structure of the differential equation, estimate x→∞limy(x). Provide a reasoned argument to support your estimate.