9-DifferntEquations Page 251 Wednesday, February 4, 2004 12:51 PM
Differential Equations and Difference Equations 251
EXHIBIT 9.1 Numerical Solutions of the Equation f ′= f with the Euler
Approximation for Different Step Sizes
This equation describes oscillatory motion, such as the elongation of a
pendulum or the displacement of a spring.
To approximate this equation we must approximate the second
derivative. This could be done, for example, by combining difference
quotients as follows:
fx( + ∆x)– fx()
f′()≈----------------------------------------
∆x
x
fx( + 2 ∆x)– fx( + ∆x)
f′(x+ ∆x) ≈---------------------------------------------------------
∆x