From the textbook, we defined ( I might be paraphrasing)

"Equations of the form $Lu = f(X) $ where Lu is a linear partial differential expression with respect to an unknown function u is called a Linear Equation. The equation is homogenous if $ f = 0 $ and inhomogenous otherwise

and used the example of $ Lu := a_{11}u_{xx} + 2a_{12}u_{xy} +a_{22}u_{yy} + a_{1}u_{x} + a_{2}u_{y} + au = f(x) $ (i) . We also know it is Non linear if it cannot be expressed in the form (i).

This leads me to my question as Problem 1 asks us to classify equations (1-10), if they are linear homogenous, inhomogenous or , non linear

and for two of them:

$ u_{t} + uu_{x} = 0 $ (2) and $ u_{t} + uu_{x} + u = 0$ (4).

Paraphrasing the text:

If $Lu = f(x) $ , and both the coefficients and right hand side expression depend on lower order derivatives, then the equation is Quasilinear

So does this mean that eqns (2) and (4) are Quasilinear then? Is that wholly seperate from the options of Linear Homogenous, Linear Inhomogenous and Nonlinear, or are Quasilinear and Semilinear sub-catagories of those?