Economics Dynamics Problems 167

# Economics Dynamics Problems 167 - Systems of rst-order...

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Systems of Frst-order differential equations 151 Figure 4.6. Consider the frst equilibrium line. Along this line we have combinations oF x and y For which ˙ x = 0. But this means that For any value oF x on this line, x cannot be changing. This inFormation is shown by the vertical dotted lines in fgure 4.6 For any particular value oF x . Similarly, For the line denoted ˙ y = 0( y = 2 x ) the value oF y on this line cannot be changing. This inFormation is shown by the horizontal dotted lines in fgure 4.6 For any particular value oF y . Next consider points either side oF the equilibrium lines in the phase plane. To the right oF the x -line we have y < x 3 implying ˙ x > 0 Hence, For any point at which x lies below the x -line, then x is rising. Two are shown in fgure 4.6 by the horizontal arrows that are pointing to the right. By the same reasoning, to the leFt oF the x -line we have y > x 3 implying ˙ x < 0 Hence, For any point at which
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