303_pdfsam_math 54 differential equation solutions odd

303_pdfsam_math 54 differential equation solutions odd -...

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Exercises 5.4 17. For critical points, we solve the system f ( x, y )=0 , g ( x, y 2 x +13 y =0 , x 2 y 2( 2 y )+13 y , x =2 y y , x . Therefore, the system has just one critical point, (0 , 0). The direction ±eld is shown in Figure B.33 in the text. From this picture we conclude that (0 , 0) is a center (stable). 19. We set v = y 0 .Then y 0 =( y 0 ) 0 = v 0 and so given equation is equivalent to the system y 0 = v, v 0 y y 0 = v 0 = y. In this system, f ( y,v )= v and g ( y . For critical points we solve f ( v , g ( y y , v and conclude that, in yv -plane, the system has only one critical point, (0 , 0). In the upper half-plane, y 0 = v> 0 and, therefore, y increases and solutions flow to the right; similarly, solutions flow to the left in the lower half-plane. See Figure B.34 in the answers of the text. The phase plane equation for the system is dv dy = dv/dx
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