I don't think it is true unless you restrict at lease some of the values of a,b and c.

To solve the complimentry solution use the ansatz

Then plut into the homogenous equation to get

factoring gives

Since the exponential function is never zero the quadratic must be zero.

Using the quadratic equation we gt

So the complimetery solution to the equation is

This will not always tend to zero as t goes to infinity. I.e if

The solution of the homoenous problem goes to infinity.

I hope this helps