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19_pdfsam_math 54 differential equation solutions odd

19_pdfsam_math 54 differential equation solutions odd -...

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Exercises 1.3 11. For this equation, the isoclines are given by 2 x = c . These are vertical lines x = c/ 2. Each element of the direction field associated with a point on x = c/ 2 has slope c . (See Figure B.7 in the answers of the text.) 13. For the equation ∂y/∂x = x/y , the isoclines are the curves x/y = c . These are lines that pass through the origin and have equations of the form y = mx , where m = 1 /c , c = 0. If we let c = 0 in x/y = c , we see that the y -axis ( x = 0) is also an isocline. Each element of the direction field associated with a point on an isocline has slope c and is, therefore, perpendicular to that isocline. Since circles have the property that at any point on the circle the 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 we cannot have y = 0 (since x/y would then have a zero in the denominator) the solutions will not be defined on the x -axis. (Note however that a related form of this differential equation is
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