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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