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Example 1.1 Find all u(x, y) satisfying the equation u xx = 0. Well, we can integrate
once to get u x = constant. But that’s not really right since there’s another variable
y. What we really get is u x (x, y) = f(y), where f(y) is arbitrary. Do it again to
get u(x, y) = f(y)x + g(y). This is the solution formula. Note that there are two
arbitrary functions in the solution. We see this as well in the next two examples.

Example 1.2 Solve the PDE u xx + u = 0. Again, it’s really an ODE with an extra
variable y. We know how to solve the ODE, so the solution is u = f(y) cos x +
g(y) sin x, where again f(y) and g(y) are two arbitrary functions of y. You can easily
check this formula by differentiating twice to verify that U xx = −u.

Example 1.3 Solve the PDE u xy = 0. This isn’t too hard either. First let’s integrate
in x, regarding y as fixed. So we get

u y (x, y) = f(y).

Next let’s integrate in y, regarding x as fixed. We get the solution

u(x, y) = F(y) + G(x),

0
where F = f.
Conclusion 1.1 A PDE has arbitrary functions in its solution. In these examples
the arbitrary functions are functions of one variable that combine to produce a func-
tion u(x, y) of two variables which is only partly arbitrary.

A function of two variables contains immensely more information than a function of only
one variable. Geometrically, it is obvious that a surface (u = f(x, y)), the graph of a
function of two variables, is a much more complicated object than a curve (u = f(x)),
the graph of a function of one variable.
To understand this book you have to know certainly all the basic facts about partial
derivatives and multiple integrals.

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