A linear partial differential equation is one of the simplest and most important classes of first-order equations. In this type, the unknown function and its first partial derivatives appear linearly. Let $u=u(x,y)$ be the unknown function. The general linear first-order partial differential equation has the form $$\begin{equation*} a(x,y)u_x+b(x,y)u_y+c(x,y)u=d(x,y). \quad \text{(1)} \end{equation*}$$ Here $a$, $b$, $c$, and $d$ are functions of the independent variables $x$ and $y$ only.

A first-order partial differential equation is called linear if it is linear in the unknown function $u$ and its first-order partial derivatives, and the coefficients depend only on the independent variables.