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equation that relates the independent variable x, the dependent variable y and derivatives
of y is called an ordinary differential equation. Some examples of ODEs are:

0
y = y
00
y + y = cos x
2 0
3 00
x y + x y − 1 = 0
000
00
0
In general, an ODE can be written as F(x, y, y , y , y , . . . ) = 0.
The key defining property of a partial differential equation (PDE) is that there is more
than one independent variable x, y, . . . There is a dependent variable that is an unknown
function of these variables u(x, y, . . .). We will often denote its derivatives by subscripts;
thus ∂u/∂x = u x , and so on. A PDE is an identity that relates the independent variables,
the dependent variable u, and the partial derivatives of u. It can be written as

F(x, y, u(x, y), u x (x, y), u y (x, y)) = F(x, y, u, u x , u y ) = O. (1.1)

This is the most general PDE in two independent variables of first order. The order of an
equation is the highest derivative that appears. The most general second-order PDE in
two independent variables is

F(x, y, u, u x , u y , u xx , u xy , u yy ) = 0. (1.2)

A solution of a PDE is a function u(x, y, . . .) that satisfies the equation identically, at least
in some region of the x, y, . . . variables. When solving an ordinary differential equation
(ODE), one sometimes reverses the roles of the independent and the dependent variables
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— for instance, for the separable ODE du = u . For PDEs, the distinction between the
dx
independent variables and the dependent variable (the unknown) is always maintained.
Some examples of PDEs (all of which occur in physical theory) are:

1) u x + u y = 0 (transport);

2) u x + yu y = 0 (transport);

3) u x + uu y = 0 (shock wave);

4) u xx + u yy = 0 (Laplace’s equation);

3
5) u tt − u xx + u = 0 (wave with interaction);
6) u t + uu x + u xxx = 0 (dispersive wave);

7) u tt + u xxxx = 0 (vibrating bar);

√
8) u t − iu xx = 0 (i = −1) (quantum mechanics).
Each of these has two independent variables, written either as x and y or as x and t.
Examples 1 to 3 have order one; 4, 5, and 8 have order two; 6 has order three; and 7 has
order four. Examples 3, 5, and 6 are distinguished from the others in that they are not
”linear.” We shall now explain this concept.

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