I'm trying to figure out how to prove the following equation in nonlinear.
(d/dx)u + u(d/dy)u = 0.
I know to show something is linear you need to show for a linear operator, L, and constants a, b and functions u, v that:
L(au + bv) = aL(u) + bL(v).
The problem I am having is that I want to show that is is a linear operator by saying that L= (d/dx) + u(d/dy). This is because I know that I can prove
(d/dx)u + x(d/dy)u = 0 is a linear operator by saying that L = (d/dx) + x(d/dy). Obviously i can use independent variables in defining a linear operator, but not use
dependent variables in defining a linear operator. Therefor, the crux of my problem is not knowing how to define the Operator to then show that the equation is nonlinear. Help on this would be appreciated. Thanks!