3.3 Analytical solution

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:

1. In one of the two ordinary differential equation solutions you obtained, say the one involving the integration constant you called , replace by the more precise to indicate that it is not really a constant, but still depends on what is.
2. Substitute for from the other ordinary differential equation solution.

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.)

Example

Question: (5.30 continued) Solve Solution:

Previously, it was found that the characteristics of this example were given by To get the general expression for , first note that more precisely, then plug in the expression for from the other equation to get This is the most general solution of the partial differential equation. Function remains 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.