Often, you want an analytical expression for the solution of a
first order equation in terms of
and
. Solving the
differential equations gives expressions valid along characteristic
lines, which is not the same thing. These expressions involve two
integration “constants”, call them
and
, that
themselves are unknown functions of
and
: if you move from one
characteristic line to another, the values of
and
will
normally change. They are only constants along the characteristic
lines.
To get a relationship for as a function of
and
, the trick
is to recognize that there is a functional dependence between the two
integration constants involved. You can use, say,
as a label
for what characteristic curve you are on: different values of
correspond to different characteristic lines. And
only
depends on what characteristic line you are on, not on the position on
the line. So
only depends on what
is;
is some
function
of
. What function that is remains unknown;
that depends on the relevant initial or boundary condition, but it is
some function.
The procedure to find as a function of
and
, or at least,
to find the most general and precise expression between these three
quantities, is therefor:
Note that in some special cases, it makes a difference in which of the
two ordinary differential equation solutions you take the integration
constant to be a function of the other one: sometimes is
not a well-defined function, but
is. (An example is in
subsubsection 3.5.5.)
ExampleQuestion: (5.30 continued) Solve
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Solution:
Previously, it was found that the characteristics of this example were given by
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To get the general expression for
, first note that more precisely,
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then plug in the expression forfrom the other equation to get
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This is the most general solution of the partial differential equation. Functionremains undetermined; the above expression is a solution of the partial differential equation regardless what one-argument function you take for
.
In fact, you need an undetermined one-argument function in the solution, because you must still match the function used to specify the relevant initial or boundary condition, also a one-argument function.