A first-order partial differential equation is one of the first ideas we study in partial differential equations. It connects an unknown function with its first partial derivatives. Let $u=u(x,y)$ be an unknown function. Here $x$ and $y$ are independent variables. The function $u$ depends on both $x$ and $y$. The general first-order partial differential equation in two independent variables is $$\begin{equation*} F(x,y,u,u_x,u_y)=0, \quad (x,y) \in D \subset \mathbb{R}^2. \quad \text{(1)} \end{equation*}$$ In (1), $F$ is a given function. The unknown part is $u$. The set $D$ is the domain where the variables $x$ and $y$ are taken.

A first-order partial differential equation is an equation which contains an unknown function of two or more independent variables and its first-order partial derivatives.