What does it mean for a term in a PDE to be linear?

Question

We define a PDE as being quasilinear if the coefficients of the highest order derivatives are linear.

What exactly does this mean though? For example, the following PDE is quasilinear:

From the definition of a quasilinear PDE, this must mean that the equations $$ 1 + \left( \frac{\partial u}{\partial y} \right)^2 \hspace{20mm} 1 + \left( \frac{\partial u}{\partial x} \right)^2 \hspace{20mm} 2 \frac{\partial u}{\partial x} \frac{\partial u}{\partial x} $$ are all linear. Why are these equations linear?

Answer 1

Quasilinear does not mean that the coefficients of the highest order are linear, it means the equation is linear with respect to the highest order derivatives. I.e. somehing like

$$\sum_{ij} a_{ij}(x, u, Du)\frac{\partial^2 u}{\partial x_i \partial x_j} + b(x, u, Du)$$