Chapter 15.1, Problem 47ES

A curve C is called a flow line of a vector field F if F is a tangent vector to C at each point along C (See the accompanying figure).

(a) Let C be a flow line for $\text{F(}x,y)=-y\text{i+}x\text{j,}$ and let $(x,y)$ be a point on C for which $y\ne 0.$ Show that the flow lines satisfy the differential equation $\frac{dy}{dx}=-\frac{x}{y}$

(b) Solve the differential equation in part (a) by separation of variables, and show that the flow lines are concentric circles centered at the origin.