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2 Basic Definitions     continued...

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.