Economics Dynamics Problems 171

# Economics Dynamics Problems 171 - Systems of ﬁrst-order...

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Unformatted text preview: Systems of ﬁrst-order differential equations 155 Figure 4.11. Figure 4.12. nonhomogeneous autonomous differential equations ˙ x = −2x − y + 9 ˙ y = −y + x + 3 ˙ The equilibrium lines in the phase plane can readily be found by setting x = 0 and ˙ y = 0. Thus ˙ x=0 ˙ y=0 implying implying y = − 2x + 9 y=x+3 which can be solved to give a ﬁxed point, an equilibrium point, namely (x∗ , y∗ ) = (2, 5). The solution equations for this system for initial condition, x0 = 2 and y0 = 2 are √ √ x(t) = 2 + 2 3 sin( 3t/2)e−(3t/2) √ √ √ y(t) = 5 − (3 cos( 3t/2)) − 3 sin( 3t/2)e−(3t/2) The equilibrium lines along with the trajectory are illustrated in ﬁgure 4.11. The analysis of this example is the same as for examples 4.4 and 4.5. In this case the ﬁxed point is at (x∗ , y∗ ) = (2, 5). The vectors of force are illustrated in ﬁgure 4.12 by the arrows. What is apparent from this ﬁgure is that the system is globally stable, and the dynamic forces are sending the system towards the ﬁxed point in a counter-clockwise motion. As we illustrated in ﬁgure 4.11, if the initial point is (x0 , y0 ) = (2, 2), then the system begins in quadrant III and tends to the ﬁxed point over time in a counter-clockwise direction, passing ﬁrst into quadrant ...
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