Classification of a PDE $$u_{xxxx}+4u_{xy}=\tan x$$
Is it linear? (I think it is linear)

And what is the order of the PDE? (it is $4$?) 

And other question:
$$u_{xx}+4u_{xyy}=0$$
the order of the last PDE is $3$ (not $2$), right?
The PDE
$u_{xxxx}+4u_{xy}=\tan x$ has order $4$, but is not linear. If $u$ and $v$ are solutions and $w:=u+v$, then  $w_{xxxx}+4w_{xy}=2 \tan x \ne \tan x$.
Hence $w$ is not a solution.
The PDE
$u_{xx}+4u_{xyy}=0$ has order $3$.
The PDE
$u_{xxxx}+4u_{xy}=\tan x$ has order $4$, but is not linear. If $u$ and $v$ are solutions and $w:=u+v$, then  $w_{xxxx}+4w_{xy}=2 \tan x \ne \tan x$.
Hence $w$ is not a solution.
The PDE
$u_{xx}+4u_{xyy}=0$ has order $3$.