A quasi-linear partial differential equation is a special type of First-Order Partial Differential Equation. In this type, the first partial derivatives appear linearly. The equation may still contain nonlinear terms in the unknown function $u$. But it must be linear in the derivatives $u_x$ and $u_y$. For an unknown function $u=u(x,y)$, the general quasi-linear first-order equation is $$\begin{equation*} a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u). \quad \text{(1)} \end{equation*}$$ Here $a$, $b$, and $c$ are functions of $x$, $y$, and $u$.

A first-order partial differential equation is called quasi-linear if it is linear in the first-order partial derivatives of the unknown function.