Now eliminating γ from the above two equations, one can obtain a closed form equation which can be used for finding the frequency response of the system.

..........................................................(6.6.37)

This shows either the system will have a trivial state response (i.e.a=0 ) and a non trivial response which can be obtained by solving the following equation.

........................................................ (6.6.38)

This equation is quadratic in and hence the solution can be written as

.....................................................................................................(6.6.39)

Where ..............................................................................................(6.6.40)

and ........................................................................(6.6.41)

As q is always positive, so the nontrivial free oscillation amplitudes occur when . For these conditions one should have

...........................................................(6.6.42)

So σ and α should have same sign. From Eq. (6.6.42), for a given nontrivial solutions can exist only if

..............................................................................................(6.6.43)

Similarly for a given σ, nontrivial solution can exist only if