Exercises 1.311.For this equation, the isoclines are given by 2x=c. These are vertical linesx=c/2. Eachelement of the direction field associated with a point onx=c/2 has slopec. (See Figure B.7in the answers of the text.)13.For the equation∂y/∂x=−x/y, the isoclines are the curves−x/y=c. These are lines thatpass through the origin and have equations of the formy=mx, wherem=−1/c,c= 0. Ifwe letc= 0 in−x/y=c, we see that they-axis (x= 0) is also an isocline. Each elementof the direction field associated with a point on an isocline has slopecand is, therefore,perpendicular to that isocline. Since circles have the property that at any point on the circlethe tangent at that point is perpendicular to a line from that point to the center of the circle,we see that the solution curves will be circles with their centers at the origin. But since wecannot havey= 0 (since−x/ywould then have a zero in the denominator) the solutions willnot be defined on thex-axis. (Note however that a related form of this differential equation is
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