MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
The differential equation dxdy=x−y is defined for all real (x,y). A particular solution passes through the point (2,1).
(a) Determine the slope of the solution curve at (2,1) and explain what this reveals about the orientation of the curve at that point.
(b) Without solving the differential equation analytically, describe whether the solution curve is increasing or decreasing for x>2 when the curve lies just below the line y=x. Provide reasoning based on the structure of the differential equation.
(c) Using the sign of dxdy in regions above and below the line y=x, justify whether the solution curve that begins at (2,1) must cross the line y=x. If it does, give an approximate x–coordinate for the point of intersection and explain your reasoning.