Prove the solution of the differential equation C(y)=0 has a given form

Prove the solution of the differential equation C(y)=0 has a given form

Let $A$ and $B$ be two constant-coefficient operators whose characteristic
polynomials have no roots in common. Then let $C=AB$. Prove that every
solution of the differential equation $C(y) = 0$ has the form $y=y_1+y_2$,
where $A(y_1)=0$ and $B(y_2)=0$.
Was thinking a proof by saying the solutions of $C$ come from a linear
combination of the solutions of $A$ and $B$, but not sure if that's valid.