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

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  • $\begingroup$ Let $F(x,u,u')=0$ be a first-order ODE for the function $u(x)$. We say that it is linear whenever it can be written as $a(x)u+b(x)u'=0$ for functions $a$ and $b$. We say that it's quasilinear if it can be written as $G(x,u)+b(x,u)u'=0$. Note that in this case, $G$ is a nonlinear part, and $b$ is a function of $x$ and $u$ instead of just $x$. This is the "one-by-one" case, i.e. first-order in a single variable $x$. The "two-by-two" case is given in the answer I posted. $\endgroup$ – Saurabh Agrawal Feb 7 '20 at 11:58
  • $\begingroup$ Check out the Chapter 3 of the book: Partial differential equations of applied mathematics, Zauder,1989 $\endgroup$ – Mark May 18 '20 at 19:07

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


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