Unformatted text preview: 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 6 = 0. If we let c = 0 in − x/y = c , we see that the yaxis ( 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...
View
Full
Document
This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Slope

Click to edit the document details