So you have two equations:

and

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?