First, we find the auxiliary equation by the quadratic.

This gives us:

Now, by using y(0)=2, we can find C1:

But

Then using the other IC, we can find C2:

Then

Therefore, the IVP in terms of a is:

Now, find the limit for the last portion of the problem. It should be kind of easy to see what a is now.

What is the limit on the right when x is unbounded?. What will give 0?