# 16 Partial differential equations

In Chapters 14 and 15 we discussed ordinary differential equations and their solutions. These are equations that contain a dependent variable *y*, which is a function of a single variable *x*, and derivatives of *y* with respect to *x*. In this chapter, we extend the discussion to similar equations that involve functions of two or more variables *x*_{1}, *x*_{2}, …, *x _{n}*. These are called

*partial differential equations (PDEs)*because the functional form analogous to (14.2) in general contains partial differentials with respect to several variables, including mixed derivatives. If we consider a function

*u*of just two variables

*x*

_{1}=

*x*and

*x*

_{2}=

*y*

*,*then examples of partial differential equations are

and

where *f*(*x*, *y*) is an arbitrary function of *x* and *y.*

By analogy with the definitions in Chapter 14, the *degree* of the equation is defined as the power to which the highest order derivative is raised after the equation is rationalised, if necessary; and the *order* of a partial differential equation is the order of the highest derivative in the equation. Thus (16.1a) is first-order and (16.1b) and (16.1c) are second-order equations. In addition, all ...

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