Examine the differential equation on the domain and .

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

Problem

Examine the differential equation dydx=x2−y2x2+y2\displaystyle \frac{dy}{dx}=\frac{x^{2}-y^{2}}{x^{2}+y^{2}} on the domain −4≤x≤4-4\le x\le4 and −4≤y≤4-4\le y\le4. (a) Determine all points in the domain where the differential equation is undefined, and explain the cause of each singularity. (b) Compute dydx\frac{dy}{dx} at each of the following twelve lattice points, avoiding any that are undefined: (−3,−3), (−3,0), (−3,3), (0,−3), (0,3), (3,−3), (3,0), (3,3), (−1,1), (1,1), (−1,−1), (1,−1)(-3,-3),\,(-3,0),\,(-3,3),\,(0,-3),\,(0,3),\,(3,-3),\,(3,0),\,(3,3),\,(-1,1),\,(1,1),\,(-1,-1),\,(1,-1). (c) On a set of axes, sketch the slope field suggested by the slopes in part (b). Clearly leave gaps wherever the differential equation is not defined. (d) Using your sketch, predict the qualitative behavior of the particular solution satisfying y(1)=0.5y(1)=0.5 as xx increases. Provide reasoning based on the local direction segments. (e) Decide whether the differential equation admits any equilibrium solutions on the given domain. Justify your conclusion analytically.

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