Difference between linear and quasi linear differential equation. Which is more strong? 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

 A: 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
$$
G(x,y,u,u_x,u_y)+d(x,y,u,u_x,u_y)u_{xy}+e(x,y,u,u_x,u_y)u_{yx}+f(x,y,u,u_x,u_y)u_{xx}+g(x,y,u,u_x,u_y)u_{yy}=0
$$
