ABSTRACT
In most partial differential equation (PDE) textbooks, first-order PDEs usually receive only a brief treatment. The theory of first-order PDEs locally reduces to the study of systems of first-order ordinary differential equation (ODE), which is presumably a subject of another course. This chapter suggests that first-order PDEs have a variety of applications. It solves first-order, linear PDEs with constant coefficients by introducing a linear change of variables, which converts the PDE into a family of ODEs depending on a parameter. The chapter handles the case of first-order, linear PDEs with nonconstant coefficients. It shows how quasi-linear PDEs arise in the study of traffic flow and nonlinear continuum mechanics, particularly with regard to the phenomenon of shock waves. In the optional, the more involved theory of arbitrary nonlinear first-order PDEs is introduced, and there is an application to the study of the motion of wave fronts in an inhomogeneous medium with a variable wave propagation speed.
