We now study characteristic equations for a first-order quasi-linear partial differential equation. These equations are ordinary differential equations. They describe the curves along which the given PDE becomes easier to understand. Consider the first-order quasi-linear equation $$\begin{equation*} a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0. \quad \text{(1)} \end{equation*}$$ From the geometrical interpretation, the vector $(a,b,c)$ gives the Characteristic Direction on the integral surface. A curve whose tangent follows this direction is called a characteristic curve.

A curve in $(x,y,u)$-space is called a characteristic curve if its tangent vector at every point coincides with the vector field $(a,b,c)$.