Unformatted text preview: has a unique solution in some neighborhood of the point a. Slope fields and geometrical solution curves are introduced in this section as a concrete aid in
visualizing solutions and existence—uniqueness questions. Solution curves corresponding to the
SIOpe fields in Problems 1—10 are shown in the answers section of the textbook and will not be duplicated here. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. Each isoclinc x— l = C is a vertical straight Iine. Each isocline x+ y = C is astraight line with slope m = —1. Eachisocline y?1 a C20, that is, y = x/E or y = —\/E, isahorizontal straight
line. Eachisocline a; = C. that is, y CG, isahoriZontal straight line. Each isocline 32/]: = C, or y = Cx, isastraight line through the origin. Each isocline x2 — y2 = C is a hyperbola that opens along the x—axis if C>0, along the
yaxis if C < 0. Each isocline xy = C isarectangular hyperbolathat opens along the line y = x if
C>O. along y = —x if C<O. Each isocline x—~ y2 = C, or y2 = x~—C, is atranslated parabola that opens along the x—axis. Each isocline y ~x2 = C, or x2 = y— C, is a translated parabola that opens along the yvaxis. X Each isocline is an exponential graph of the form y = Ce . Because both f (x, y) = Early2 and 8f fay = 4x2)! are continuous everywhere, the
existence~uniqueness theorem of Section 1.3 in the textbook guarantees the existence of a
unique solution in some neighborhood of x = 1. Both f(x,y) 2' xlny and Bf/ay = x/y are continuous in aneighborhood of
(l, 1), so the theorem guarantees the existence of a unique solution in some
neighborhood of x = 1. Both f(x,y) = y”3 and {if/By : (113))?” are continuous near (0,1), so the
theorem guarantees the existence of a unique solution in some neighborhood of x = 0. Section 1.3 11 ...
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