A PDE in **u** is classified as linear if all
of the terms involving **u** and any of its derivatives can be
expressed as a linear combination in which the coefficients
of the **u**-terms are independent of **u**. In a linear PDE, the
coefficients can depend at most on the independent
variables.

If is a function of the two
independent variables **x** and **y**, then

is a linear (constant coefficient) PDE.

The PDE

is a linear (variable coefficient) PDE.

The PDE

is nonlinear.