14 Ordinary differential equations

Any equation that expresses the functional dependence of a variable y on its arguments xi(i = 1, 2, …) and the derivatives of y with respect to those arguments, is called a differential equation. Physical systems are almost always described by such equations. For example, a wave f(x, t) travelling with velocity in the x-direction obeys the wave equation

(14.1a) Unnumbered Display Equation

while the motion of a simple pendulum of length l performing small oscillations θ satisfies

(14.1b) Unnumbered Display Equation

where g is a constant. Equation (14.1a) contains partial derivatives because f is a function of more than one variable, and the equation is therefore called a partial differential equation (PDE). These will be discussed in Chapter 16. On the other hand, in Equation (14.1b) the quantity θ depends on the single variable t, and the equation is called an ordinary differential equation (ODE). It is these equations that are the subject of this chapter and the next.

In what follows, we will usually refer to the independent variable as x and the corresponding dependent variable as y(x). Thus, in general, an ordinary differential equation is of the form


Examples of ODEs are:

(14.3a) ...

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