ya.. the differential eqn. for shm is

 

accl^n + (angular velocity)².displacement = 0 (avoiding symbols for they are already used)

 

now substract the two equations .....(adding makes LHS zero...but we need d²x/dt² in our eqn.) 

we get..

2m.d²x/dt² = -2m²x - 4kx

or 2.d²x/dt² = -2²x - 4(k/m)x

or d²x/dt² = -²x - 2(k/m)x

or d²x/dt² + ²x + 2(k/m)x = 0

or d²x/dt² + {² + 2(k/m)}.x = 0

now for the combined system... disp. is 2x

so multiply 2 in both sides..

we get..

d²(2x)/dt² + {² + 2(k/m)}.2x = 0

 

so we'r getting an eqn. of the form

accl^n + (angular velocity)².displacement = 0

 

So that term present as a product of the disp. 2x is the angular velocity here..

 

So ( ^/ )² = ² + 2(k/m)

So Time period = 2/(  ^/ )... put the value there...

So T = 2/ { ² + 2(k/m) }

        = 2/ { (g/L) + 2(k/m) }