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Unformatted text preview: , 0). Among pictures shown in ±igure 5.7, only the unstable node and the unstable spiral have this feature. Since the unstable node is the answer to (a), we have the unstable spiral in this case. (e) The phase plane equation dy dx = 4 x − 3 y 5 x − 3 y , has two linear solutions, y = 2 x and y = 2 x/ 3. (One can Fnd them by substituting y = ax into the above phase plane equation and solving for a .) Solutions starting from a point on y = 2 x in the Frst quadrant, have x = 5 x − 3(2 x ) = − x < 0 and so ﬂow toward (0 , 0); similarly, solutions, starting from a point on this line in the third quadrant, have x = − x > 0 and, again, ﬂow to (0 , 0). On the other line, y = 2 x/ 3, the picture is opposite: in the Frst quadrant, x = 5 x − 3(2 x/ 3) = 3 x > 0, and x < 0 in the third quadrant. Therefore, there are two lines, passing through the critical point (0 , 0), such 302...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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