342_pdfsam_math 54 differential equation solutions odd

342_pdfsam_math 54 differential equation solutions odd -...

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Chapter 5 x 4 = x 5 , x 5 = x 6 , x 6 = x 2 x 3 . 13. With the notation used in (1) on page 264 of the text, f ( x, y ) = 4 4 y, g ( x, y ) = 4 x, and the phase plane equation (see equation (2) on page 265 of the text) can be written as dy dx = g ( x, y ) f ( x, y ) = 4 x 4 4 y = x y 1 . This equation is separable. Separating variables yields ( y 1) dy = x dx ( y 1) dy = x dx ( y 1) 2 + C = x 2 or x 2 ( y 1) 2 = C , where C is an arbitrary constant. We find the critical points by solving the system f ( x, y ) = 4 4 y = 0 , g ( x, y ) = 4 x = 0 y = 1 , x = 0 . So, (0 , 1) is the unique critical point. For y > 1, dx dt = 4(1 y ) < 0 , which implies that trajectories flow to the left. Similarly, for
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Unformatted text preview: y < 1, trajectories flow to the right. Comparing the phase plane diagram with those given on ±igure 5.12 on page 270 of the text, we conclude that the critical point (0 , 1) is a saddle (unstable) point. 15. Some integral curves and the direction Feld for the given system are shown in ±igure 5-B. Comparing this picture with ±igure 5.12 on page 270 of the text, we conclude that the origin is an asymptotically stable spiral point. 338...
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