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# 0011 - has a unique solution in some neighborhood of the...

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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 y-axis 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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