= Solution
Multiply the supplied <inhour equation> by $\omega+\lambda$ and collect powers of the inverse period:
$$
\Lambda\omega^2+(\Lambda\lambda+\beta-\rho)\omega-\rho\lambda=0.
$$
With the stipulated constants this becomes
$$
\omega^2+12.1\omega-0.2=0,
$$
with roots $\omega_+=0.0165064\ \mathrm{s}^{-1}$ and $\omega_-=-12.1165\ \mathrm{s}^{-1}$. The negative root describes a rapidly decaying component. The positive root governs the late-time growth, giving
$$
\boxed{T_+=\omega_+^{-1}=60.58\ \mathrm s\simeq60.6\ \mathrm s.}
$$
The growing-mode e-folding time is not the precursor mean lifetime, and the negative root is a legitimate transient mode rather than an unphysical negative time.
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