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Unformatted text preview: Systems of ﬁrst-order differential equations 183
Figure 4.31. illustrate a stable limit cycle, but they also illustrate that the limit cycle shrinks as
β increases. 4.11 Euler’s approximation and differential equations
on a spreadsheet10
Although differential equations are for continuous time, if our main interest is the
trajectory of a system over time, sometimes it is convenient to use a spreadsheet
to do this. To accomplish this task we employ Euler’s approximation. For a single
variable the situation is shown in ﬁgure 4.32. We have the differential equation
= f (x, t)
x(t0 ) = x0
Let x = φ (t) denote the unknown solution curve. At time t0 we know x0 = φ (t0 ).
We also know dx/dt at t0 , which is simply f (x0 , t0 ). If we knew x = φ (t), then the
value at time t1 would be φ (t1 ). But if we do not have an explicit form for x = φ (t),
we can still plot φ (t) by noting that at time t0 the slope at point P is f (x0 , t0 ), which
is given by the differential equation. The value of x1 at time t1 (point R) is given
x1 = x0 + f (x0 , t0 ) t (4.30) t = t1 − t0 This process can be repeated for as many steps as one wishes. If f is autonomous,
so dx/dt = f (x), then
xn = xn−1 + f (xn−1 ) t
It is clear from ﬁgure 4.32 that point R will deviate from its ‘true’ value at
point Q, the larger the step size, given by t. If the step size is reduced, then the
approximation is better.
10 See Shone (2001) for a treatment of differential equations with spreadsheets. (4.31) ...
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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