chapter2.page19

chapter2.page19 - 5 PHASE AND VECTOR FIELD DIAGRAMS 19...

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Unformatted text preview: 5. PHASE AND VECTOR FIELD DIAGRAMS 19 Figure 12. Vector field for f (t, x) = sin(t) of the dynamics of solutions as shown in the next not so complicated diagram, Figure 13. Vector field for f (t, x) = t - x2 To graph the vector field diagram for y = f (x, y) by hand over rectangular region [a, b] [c, d], you would do the following: (1) Divide interval [a, b] into N equal length subintervals and [c, d] into M equal length subintervals, for [a, b] the length is h = b-a and the subintervals are [a, a+h], [a+h, a+2h], ; N for [c, d] the subinterval length is s = d-c and the subintervals M are [c, c+s], [c+s, c+2s], . (2) If we use xi = a + ih, i = 0, 1, , N to denote the endings of subintervals of [a, b] and yj = c + js, j = 0, 1, 2, to denote the endings of subintervals of [c, d], then (xi , yj ) form the grid points, i = 0, 1, , N ; j = 0, 1, , M. For each point (xi , yj ) compute the slope mij = f (xi , yj ) and, from the ...
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