The homogoneous DE is $Ly=0$, where $L$ is a linear differential operator, such as (for example)
Observe that for the above operator, if $u,v$ are functions and $\alpha,\beta$ constants, we have
$$L(\alpha u+\beta v)=\alpha Lu+\beta Lv.$$
This is what it means to be linear. We are given the following two facts ($b$ is a function):
$$Ly_1=b, \quad Ly_2=0.$$
Note $b$ is not the zero function and $L$ is now arbitrary. Using these, the four questions become:
- Does $L(-y_1)=b$?
- Does $L(-y_2)=0$?
- Does $L(y_1-y_2)=b$?
- Does $L(y_2-y_1)=0$?
Using the linearity properties of $L$, reduce the above expressions to expressions in just $Ly_1$ or $Ly_2$, then replace these respectively with $b$ and $0$, and then decide if the resulting equation is true/false.