Page 240 - Excel for Scientists and Engineers: Numerical Methods

Chapter 10

Numerical Integration of

Ordinary Differential Equations

Part I: Initial Conditions

A differential equation is an equation that involves one or more derivatives.
Many physical problems, when formulated mathematically, lead to differential
equations. For example, the equation (k > 0)

-=- kY (10-1)
dy
dt
describing the decrease in y as a function of time, occurs in the fields of reaction
kinetics, radiochemistry or electrical engineering (where y represents
concentration of a chemical species, or atoms of a radioactive element, or
electrical charge, respectively) as well as in many other fields. Of course, a
differential equation can be more complicated that the one shown in equation 10-
1 ; another example from electrical engineering is shown in equation 10-2,
di
L- + Ri = E (1 0-2)
dt

where R is the resistance in a circuit, L is the inductance, E is the applied
potential, i is the current and t is time.

If a differential equation contains derivatives of a single independent
variable, it is termed an ordinary differential equation (ODE), while an equation
containing derivatives of more than one independent variable is called a partial
differential equation (PDE). Partial differential equations are discussed in a
subsequent chapter.
The general form of an ordinary differential equation is

(1 0-3)

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