Classification of first order PDEs

To begin with, we are interested in first order PDE in two unknowns. The most generic form of such equation is

$$F(x,y,u,u_x,u_y)=0,$$

where $x$ and $y$ are independent variables, $u$ is dependent variable. This first order PDEs are classified as:

1. A PDE is said to be linear if the dependent variable $u$ and its partial derivative ($u_x$ and $u_y$) occur only in the first degree and are not multiplied. e.g.
   $$a(x,y)u_x+b(x,y)u_y=c_1(x,y)u +c_2(x,y).$$

2. When $F$ is not a linear function of $u_x$ and $u_y$, then it is said to be non-linear PDE.

3. When $F$ is a linear function of $u_x$ and $u_y$, but not necessarily linear $u$ then it is called quasi-linear equation. e.g.
   $$a(x,y,u)u_x+b(x,y)u_y=c(x,y,u).$$

4. The coefficients of $u_x$ and $u_y$ do not depend on $u,$ and the nonlinear in the equation is present only in the inhomogeneous term, then the equation is called semi-linear equation. e.g.
   $$a(x,y)u_x+b(x,y)u_y=c(x,y,u).$$