# 2.0: Prelude to First Order Equations

For a given sufficiently regular function \(F\) the general equation of first order for the unknown function \()\) is

$$F(x,u,\nabla u)=0$$

in \(n\). The main tool for studying related problems is the theory of ordinary differential equations. This is quite different for systems of partial differential of first order. The general linear partial differential equation of first order can be written as

$$\sum_{i=1}^na_i(x)u_{x_i}+c(x)u=f(x)$$

for given functions \(a_i,\ c\) and \(f\). The general *quasilinear *partial differential equation of first order is

$$\sum_{i=1}^na_i(x,u)u_{x_i}+c(x,u)=0.$$