The study of Partial Differential Equations (PDEs) is a cornerstone of advanced mathematics, providing the language to describe physical phenomena ranging from fluid dynamics to quantum mechanics. A first-order PDE is an equation involving an unknown function $u(x, y, \dots)$ and its first-order partial derivatives. In the case of two independent variables $x$ and $y$, the general form is: $$\begin{equation} F(x, y, u, u_x, u_y) = 0 \cdots\text{(1.1)}\nonumber \end{equation}$$ To simplify notation in undergraduate lectures, we frequently use the standard symbols $p$ and $q$ to represent the partial derivatives: $$\begin{equation} p = \frac{\partial u}{\partial x}, \quad q = \frac{\partial u}{\partial y} \nonumber \end{equation}$$ Understanding whether an equation is linear or nonlinear is the most critical step in choosing a solution strategy, such as Lagrange's method or the method of characteristics.

A first-order partial differential equation is defined as linear if it is linear with respect to the unknown function $u$ and its partial derivatives $u_x$ and $u_y$. The general mathematical structure is: $$\begin{equation} a(x, y)u_x + b(x, y)u_y + c(x, y)u = d(x, y) \cdots\text{(1.2)}\nonumber \end{equation}$$ where the coefficients $a, b, c$ and the term $d$ are functions of the independent variables $x$ and $y$ only.