Remember that if you have an equation p(x)=0 where p(x) is a polynomial, and you factor p into (x-a)(x-b)... then the roots are a,b,.... Here you're doing the same thing backwards.
Once you get the characteristic equation, the D.E. should be easy to find. You would normally have the D.E. and write the characteristic equation, but either way is easy.
If you can't get the answer, let us know where you get stuck.