I am currently studying PDE for the first time. So I came across some definitions of linear differential operator and quasi-linear differential operator. What exactly is the difference? Can someone explain in simple words?
This is the definition in my script
It's helpful to consider the "two by two" case, i.e. second-order PDEs in $x$ and $y$. Recall that such a PDE always has the form $$F(x,y,u,u_x,u_y,u_{xy},u_{yx},u_{xx},u_{yy})=0$$ for some function $F$. This PDE is called linear whenever it can be written $$ a(x,y)u+b(x,y)u_x+c(x,y)u_y+d(x,y)u_{xy}+e(x,y)u_{yx}+f(x,y)u_{xx}+g(x,y)u_{yy}=0 $$ which we will abbreviate $$ au+bu_x+cu_y+du_{xy}+eu_{yx}+fu_{xx}+gu_{yy}=0 $$ The quasilinear case is different in two respects. First of all, the original function $F$ need only be linear in derivatives of the highest order (in this case $\text{ord}=2$). That is to say, it can be written as $$ G(x,y,u,u_x,u_y)+du_{xy}+eu_{yx}+fu_{xx}+gu_{yy}=0 $$ for some (possibly nonlinear) function $G$. The second difference is this: the remaining coefficients of the "linear part" (or principal part) can depend on $u$, $u_x$, and $u_y$ as well as $x$ and $y$. In particular, we should write $$ G(x,y,u,u_x,u_y)+d(x,y,u,u_x,u_y)u_{xy}+e(x,y,u,u_x,u_y)u_{yx}+f(x,y,u,u_x,u_y)u_{xx}+g(x,y,u,u_x,u_y)u_{yy}=0 $$