# 14 Ordinary differential equations

Any equation that expresses the functional dependence of a variable *y* on its arguments *x _{i}*(

*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

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

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

(14.2)

Examples of ODEs are:

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