So you have two equations:
Take (2) and solve for thus:
Differentiate this equation twice:
We also differentiate (1) twice thus:
Now plug (3) and (4) into (5) thus:
This equation can, most likely, be massaged to fit your characteristic equation. If not, then try solving (1) for differentiating twice, and substituting into the twice-differentiated version of (2). You'll have to see how your and fit into that scheme.
Does that get you started?