Page 321 - Schaum's Outline of Differential Equations

CHAPTER 31

An Introduction to

Partial Differential

Equations

INTRODUCTORY CONCEPTS
A partial differential equation (PDE) is a differential equation in which the unknown function depends on
two or more independent variables (see Chapter 1). For example,

is a PDE in which u is the (unknown) dependent variable, while x and y are the independent variables. The def-
initions of order and linearity are exactly the same as in the ODE case (see Chapters 1 and 8) with the proviso
that we classify a PDE as quasi-linear if the highest-order derivatives are linear, but not all lower derivatives
are linear. Thus, Eq. (31.1) is a first-order, linear PDE, while

is a second-order, quasi-linear PDE due to the term.

Partial differential equations have many applications, and some are designated as classical, much like their ODE
counterparts (see Chapter 29). Three such equations are the heat equation

the wave equation

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