How to decide whether PDE is Homogeneous or non-homogeneous. I am studying second order PDE.
And I have seen homogeneous and non-homogeneous PDE. 
But I cannot decide which one is homogeneous or non-homogeneous. 
For examples;
(1)  $(D^3-3D^2D'+4D'^3)u=0$ 
The equation is said to be homogeneous. Why? 
(2) $x^2u_{xx}-y^2u_{yy}2xu_x+2yu_y=0$
Homogeneous or nonhomogeneous? Why? 
(3) $x^2u_{xx}-y^2u_{yy}=xy$
Homogeneous or nonhomogeneous? Why? 
I wrote some examples just because you can explain more efficiently. Thanks a lot. 
 A: Imagine that $u$ solves the PDE and check whether every function $\alpha u$ solves it too. If they do, the PDE is homogeneous, otherwise it is not.
The method is quite easy and short. In case (2) for example, the LHS for $\alpha u$ becomes
$$\alpha x^2u_{xx}-\alpha^2y^2u_{yy}2xu_x+2\alpha yu_y=\alpha (x^2u_{xx}-y^2u_{yy}2xu_x+2yu_y)+(\alpha-\alpha^2)y^2u_{yy}2xu_x,
$$
which is, if $u$ solves the PDE and for every $\alpha$ not $0$ or $1$, a nonzero multiple of
$$
y^2u_{yy}2xu_x,
$$
not always zero, hence the PDE is not homogeneous. Likewise, the LHS of (3) becomes 
$$
\alpha(x^2u_{xx}-y^2u_{yy}),
$$
hence, if $u$ solves the PDE, $\alpha u$ solves the PDE if, for every $(x,y)$,
$$
\alpha xy=xy.
$$
This is obviously false hence (3) is not homogeneous. And so on.
A: Eqs. (1) and (2) are of the form
$$
\mathcal{D} u = 0
$$
where $\mathcal D$ is a differential operator. Thus, these differential equations are homogeneous. Eq. (3), of the form
$$
\mathcal{D} u = f \neq 0
$$
is non-homogeneous. The definition of homogeneity as a multiplicative scaling in @Did's answer isn't very common in the context of PDE. However, it works at least for linear differential operators $\mathcal D$. See also this post.
