Given two solutions to a nonhomogenous equation, $y_{{p}_{1}}(t)$ and $y_{{p}_{2}}(t)$, then the sum $y_{{p}_{1}}(t)+y_{{p}_{2}}(t)$ is also a particular solution.
I know it's false from a counterexample, but for the assignment, our professor doesn't allow us to use counterexamples to show why it's false but rather wants us to evaluate the statements more generally. How would I do this?
One non-example argument would be
$$L(y_{p_1}+y_{p_2})=L(y_{p_1})+L(y_{p_2})=f+f=2f \neq f$$
when the equation is $Ly=f$.