es(a)The solution of the differential equation is a family

Chapter 2, Problem 46E

(choose chapter or problem)

Streamlines

                               

(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, \(\pm 0.2\), \(\pm 0.4\), \(\pm 0.6\), and \(\pm 0.8\) in three different ways. First, use the contour plot 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.

Text Transcription:

2xy/(x^2 + y^2)^2 dx + [1 + y^2 - x^2/(x^2 + y^2)^2] dy = 0

x^2 + y^2 = 1

pm 0.2

pm 0.4

pm 0.6

pm 0.8