Particular integral of the following PDE is coming out to be different if I apply different rules: $$(D^2 +3DD'+2D'^2)z = x+y$$
where $D = \dfrac{\partial}{\partial x}$, $D' = \dfrac{\partial}{\partial y}$
Method 1: $$F(D,D')z = f(ax+by)$$ here $F(a,b) \neq 0$, so $P.I. = \frac{(x+y)^3}{6}$
Method 2: By taking $$ 2D'^2 $$ common in the denominator, and expanding the denominator in form of $$ (1 + p)^{-1}. $$ I am getting $$ \frac{xy^2}{4} - \frac{y^3}{24}. $$
Method 3 Similarly by taking $D^2$ common, I am getting a different value.
Can the above methods be applied to this PDE ? If yes, then why is the answer different for different methods?