Economics Dynamics Problems 213

Economics Dynamics Problems 213 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations If the equations for x(t), y(t) and z(t) are already known, then we use the spacecurve command, as illustrated in the following instructions: traj=spacecurve([exp(t),2*exp(t)-exp(2*t)+2*t*exp(t), 4*t*exp(t)-exp(2*t)+3*exp(t)], t=0..5,labels=[x,y,z]); Exercises 1. 2. (i) Show that y y (x) = 2x is a separable function, and solve assuming x(0) = 2 and y(0) = 3. (ii) Verify your result using either Mathematica or Maple. For the system ˙ x = x − 3y ˙ y = −2x + y use a software package to derive the trajectories of the system for the following initial values: (a) (b) (c) (d) 3. (x0 , y0 ) = (4, 2) (x0 , y0 ) = (4, 5) (x0 , y0 ) = (−4, −2) (x0 , y0 ) = (−4, 5) For the system ˙ x = −3x + y ˙ y = x − 3y (i) Show that points (x0 , y0 ) = (4, 8) and (x0 , y0 ) = (4, 2) remain in quadrant I, as in figure 4.9. (ii) Show that points (x0 , y0 ) = (−4, −8) and (x0 , y0 ) = (−4, −2) remain in quadrant III, as in figure 4.9. (iii) Show that points (x0 , y0 ) = (2, 10) and (x0 , y0 ) = (−2, −10) pass from one quadrant into another before converging on equilibrium. (iv) Does the initial point (x0 , y0 ) = (2, −5) have a trajectory which converges on the fixed point without passing into another quadrant? 4. For the system ˙ x = −2x − y + 9 ˙ y = −y + x + 3 5. establish the trajectories for each of the following initial points (i) (x0 , y0 ) = (1, 3), (ii) (x0 , y0 ) = (2, 8), and (iii) (x0 , y0 ) = (3, 1), showing that all trajectories follow a counter-clockwise spiral towards the fixed point. Given the dynamic system ˙ x = 2x + 3y ˙ y = 3x + 2y 197 ...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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