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 28(choose chapter or problem)
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. Suppose that .x.t/;y.t// is a solution of the autonomous system and that , 6D 0. Dene /.t/ D x.t C ,/ and .t/ D y.t C,/. Then show (in contrast with the situation in 27) that ./.t/; .t// is also a solution of the system. Thus autonomous systems have the simple but important property that a t-translate of a solution is again a solution.
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