What am I doing in the following? Is there a name for it? Please be polite.
Say we know, for linear differential operators $D_1,D_2$ that the following two equations are true
$$\,\,(D_1\circ D_2)f(x)=0 \tag{1}$$ $$\quad\quad\quad\quad \quad\quad(D_1\circ D_2)f(x)=(D_2\circ D_1)f(x) \tag{2}$$ So that $D_1$ and $D_2$ commute.
We then say that $f(x)$ can be written as $$f(x)=f_1(x)+f_2(x) \tag{3}$$ where $$D_1f_1(x)=0 \tag{4}$$ $$D_2f_2(x)=0 \tag{5}$$
This appears true because substituting (3) into the left-hand side of (1), we actually satisfy (1), we have \begin{align} (D_1\circ D_2) (f_1(x)+f_2(x) )&=(D_1\circ D_2) f_1(x)+(D_1\circ D_2) f_2(x) \\ &=(D_2\circ D_1) f_1(x)+(D_1\circ D_2) f_2(x) \\ &=0+0 \end{align} Where I have used first, linearity, then commutativity, then (4) with (5) in the last step. Is there a name for what's going on here?
Can the solutions of (1) always be split up as in (3)?
Other Information.
The material of this question is relevant to fluid mechanics, to the theory of the instability of jets.