This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Systems of ﬁrstorder 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 counterclockwise spiral towards the ﬁxed
point.
Given the dynamic system
˙
x = 2x + 3y
˙
y = 3x + 2y 197 ...
View
Full
Document
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.
 Fall '11
 Dr.Gwartney
 Economics

Click to edit the document details