Linear quasi-linear and nonlinear partial differential equations form one of the first structural classifications in the subject. This classification depends on how the unknown function $u$ and its partial derivatives occur in the equation. We shall define these three types carefully and illustrate them through basic second-order examples.

A partial differential equation is called linear if it is linear in the unknown function and in all the partial derivatives of the unknown function, with coefficients depending only on the independent variables. Thus, a linear partial differential equation contains no products, powers, or nonlinear functions of $u$ and its derivatives. For instance, terms such as $u^2$, $u_xu_y$, $uu_{xx}$, and $\sin u$ do not occur in a linear partial differential equation.