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Unformatted text preview: Systems of ﬁrst-order differential equations 185 Hence,
x1 = x0 + f (x0 , y0 ) t = 2 + 3(0.01) = 2.03
y1 = y0 + g(x0 , y0 ) t = 2 + 3(0.01) = 2.03
f (x1 , y1 ) = −2(2.03) − 2.03 + 9 = 2.91
g(x1 , y1 ) = −2.03 + 2.03 + 3 = 3
x2 = x1 + f (x1 , y1 ) t = 2.03 + 2.91(0.01) = 2.0591
y2 = y1 + g(x1 , y1 ) t = 2.03 + 3(0.01) = 2.06
This process is repeated. But all this can readily be set out on a spreadsheet, as
shown in ﬁgure 4.33.
The ﬁrst two columns are simply the differential equations. Columns (3) and
(4) employ the Euler approximation using relative addresses and the absolute
address for t. The x-y plot gives the trajectory of the system in the phase
plane, with initial value (x0 , y0 ) = (2, 2). As can be seen from the embedded
graph in the spreadsheet, this trajectory is the same as that shown in ﬁgure 4.11
The advantage of using Euler’s approximation, along with a spreadsheet, is that
no explicit solution need be obtained – assuming that one exists. By reducing the
step size a smoother trajectory results. It is also easy to increase the number of
steps. Figure 4.33. ...
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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.
- Fall '11