Definition of a linear differential equation

[Niles]

Homework Statement

Hi all. Is the following definition of a partial linear differential equation correct?


A partial differential equation is called linear if no products of the function and/or its derivatives occur, and if the unknown function and its partial derivatives occur only to the first degree.


I am a little worried about the "and/or" regarding products in the DE. Is it correct that the following term is NOT allowed in a linear differential equation?:

$$\frac{\partial u}{\partial x}\frac{\partial u}{\partial t}$$

- and can you confirm the validity of the "and/or" in my definition? Thanks in advance.

sincerely,
Niles.

 

[Niles]

Can anybody confirm this?

And also: If I have a function f(x,y)*u, then is the equation still homogeneous and linear? As an example, take f(x,y)=cos(x):