Consider the function
where
is an arbitrary function. It is easier to understand what is happening if we define
The partial derivatives, on using the chain rule, are
and
Therefore
for any function
This illustrates an important point. Ordinary differential equations have arbitrary constants but partial differential equations have arbitrary
functions. The functionis determined by an initial condition.
Assume that the initial condition, at
is
witha given function. If
then
Hence, the unknown function,is the given functionbut the argument is replaced by
hence, the solution is
If
then 
satisfies the equation and the initial condition.
Example: Consider the linear equation
The solution is
whereis an arbitrary function. The partial derivatives are
and
Hence
and the equation is satisfied.
Example: Consider the linear equation with non-constant coefficients,
The solution is given by
whereis again an arbitrary function. Denoting
and
and so the equation is clearly satisfied.
The argument of the function f is determined using the method of characteristics.