28 through 30 deal with the systemdx dt D F.x;y/;dy dt D G.x;y/in a region where the

Chapter 6, Problem 29

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28 through 30 deal with the systemdx dt D F.x;y/;dy dt D G.x;y/in a region where the functions F and G are continuously differentiable, so for each number a and point .x0;y 0/, there is a unique solution withx.a/D x0 and y.a/D y0. Let .x1.t/;y1.t// and .x2.t/;y2.t// be two solutions having trajectories that meet at the point .x0;y0/; thus x1.a/ D x2.b/ D x0 and y1.a/ D y2.b/ D y0 for some values a and b of t. Dene x3.t/ D x2.t C,/ and y3.t/ D y2.t C,/; where , D b!a, so.x2.t/;y2.t// and .x3.t/;y3.t// have the same trajectory. Apply the uniqueness theorem to show that .x1.t/;y1.t// and .x3.t/;y3.t// are identical solutions. Hence the original two trajectories are identical. Thus no two different trajectories of an autonomous system can intersect.