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4

Numerical Differentiation, Integration, and
Solutions of Ordinary Differential Equations

This chapter discusses the basic methods for numerically finding the value of
the limit of an indeterminate form, the value of a derivative, the value of a
convergent infinite sum, and the value of a definite integral. Using an
improved form of the differentiator, we also present first-order iterator tech-
niques for solving ordinary first-order and second-order linear differential
equations. The Runge-Kutta technique for solving ordinary differential equa-
tions (ODE) is briefly discussed. The mode of use of some of the MATLAB
packages to perform each of the previous tasks is also described in each
instance of interest.

4.1 Limits of Indeterminate Forms

DEFINITION If lim ( )ux = lim ( )v x = 0 , the quotient u(x)/v(x) is said to have
x x
x→ 0 x→ 0
an indeterminate form of the 0/0 kind.
• If lim ( )ux = lim ( )v x = ∞ , the quotient u(x)/v(x) is said to have an
x x
x→ 0 x→ 0
indeterminate form of the ∞/∞ kind.
In your elementary calculus course, you learned that the standard tech-
nique for solving this kind of problem is through the use of L’Hopital’s Rule,
which states that:

′
()
ux
if: lim = C (4.1)
′
x vx
x→ 0 ()
()
ux
then: lim = C (4.2)
x vx
x→ 0 ()
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© 2000 by CRC Press LLC
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