We study the integral surface of a first-order PDE from a geometrical point of view. The main idea is simple. A solution $u=u(x,y)$ can be seen as a surface in three-dimensional space. The given partial differential equation then tells us a direction that must lie on this surface. Consider the first-order quasi-linear equation $$\begin{equation*} a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0. \quad \text{(1)} \end{equation*}$$ Here $x$ and $y$ are independent variables, and $u$ is the dependent variable.

A surface in $(x,y,u)$-space is called an integral surface of (1) if it represents a solution $u=u(x,y)$ of the given partial differential equation.