Suppose that $y_1$ is a solution to $Ly_1 = f(x)$ and $y_2$ is a solution to $Ly_2 = g(x)$
Show that $y = y_1 + y_2$ solves $Ly = f(x) + g(x)$.
So, assuming that the linear operator $L$ is the same for both: $$\frac {Ly_1}{f(x)}=1=\frac {Ly_2}{g(x)}$$ Then, setting them equal to each other: $$\frac {Ly_1}{f(x)}=\frac {Ly_2}{g(x)}$$ Therefore: $$Ly_1g(x)=Ly_2f(x)$$ But I don't know how to go about proving the principle.