In this chapter, we discuss the solution of first-order differential equations. The main technique that is used is known as the method of characteristics, in which various paths in the domain of the independent variables, known as characteristics, are obtained. The solutions are then propagated through these paths, yielding the characteristic solution curves. (Note that the two terms are different: characterics vs. characteristic curves). A solution surface can then be constructed by combining (or “bundling”) these characteristic curves.
The method of characteristics is applicable to partial differential equations that take on particular forms known as quasilinear equations. Special cases are the semi-linear, linear, and strictly linear forms. It can be shown that for the semilinear cases, the characteristics will not intersect each other. However, for the general quasi-linear equations, the characteristics can intersect. When they do, the solution will become discontinuous, and the discontinuities are known as shocks. Moreover, if the discontinuity is given in the initial conditions, rarefaction occurs, in which a fan of characteristics are filled in to complete the solution surface. A brief treatment of shocks and rarefaction is given in Section J.1 as an appendix.
In Section 10.3, we discuss a set of conditions known as Lagrange-Charpit conditions. These conditions are used to generate the solutions of some important classes of nonlinear first order PDEs.