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.
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.