Consider the differential equation

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Consider the differential equation dydx=sin⁡x−y2\dfrac{dy}{dx}=\sin x-\dfrac{y}{2}. A slope field for this equation is provided. Selected slope values at certain lattice points are given in the table below. (Point , Approximate slope) (0,0):    0(0,0):\;\;0  (π2,0):    1(\tfrac{\pi}{2},0):\;\;1  (π,0):    0(\pi,0):\;\;0 (0,1):    −0.5(0,1):\;\;-0.5 (π2,1):    0.5(\tfrac{\pi}{2},1):\;\;0.5 (π,1):    −0.5(\pi,1):\;\;-0.5 (0,2):    −1(0,2):\;\;-1  (π2,2):    0(\tfrac{\pi}{2},2):\;\;0  (π,2):    −1(\pi,2):\;\;-1 Use the slope information to analyze behavior of solution curves. (a) Identify all listed points at which the slope of a solution curve is positive. Explain your answer by referencing the table. (b) For a solution curve passing through (π2,0)(\tfrac{\pi}{2},0), describe whether the curve is increasing or decreasing immediately to the right of x=π2x=\tfrac{\pi}{2}. Justify your conclusion using the slope field data. (c) Explain why any solution curve that intersects the line y=0y=0 at x=πx=\pi must have a local extremum there. Indicate whether it is a local maximum or minimum and justify using slope signs on each side of the point.

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