(a) The solution of the differential equation 2xy (x2 y2 ) 2 dx c1 y2 2 x2 (x2 y2 ) 2 d

Chapter 2, Problem 46

(choose chapter or problem)

(a) The solution of the differential equation

         \(\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}} d x+\left[1+\frac{y^{2}-x^{2}}{\left(x^{2}+y^{2}\right)^{2}}\right] d y=0\)

is a family of curves that can be interpreted as streamlines of a fluid flow around a circular object whose boundary is described by the equation \(x^{2}+y^{2}=1\). Solve this DE and note the solution f (x, y) = c for c = 0.

(b) Use a CAS to plot the streamlines for \(c=0, \pm 0.2, \pm 0.4, \pm 0.6, \text { and } \pm 0.8\) in three different ways. First, use the contourplot of a CAS. Second, solve for x in terms of the variable y. Plot the resulting two functions of y for the given values of c, and then combine the graphs. Third, use the CAS to solve a cubic equation for y in terms of x.