  From the principles of quantum cosmologies we can justify the reason for an
inverse-square law for the cosmological constant with no conflict with
observations. Although this general expression for $\Lambda$ is well known in
the literature, in this work we introduce some analytical solutions for the
scale factor different from the literature. The underlying motivation to carry
out these solutions is the fact that the time variation of $\Lambda$ can lead
to a creation of matter and/or radiation such as to help in the investigation
of the age of the Universe. The knowledge of the scale factor behavior might
shed some light on these questions since the entire evolution of a homogeneous
isotropic Universe is contained in the scale factor.
