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We consider the application of group invariant transformations in order toconstrain a flat isotropic and homogeneous cosmological model, containing of aBrans-Dicke scalar field and a perfect fluid with a constant equation of stateparameter $w$, where the latter is not interacting with the scalar field in thegravitational action integral. The requirement that the Wheeler-DeWitt equationbe invariant under one-parameter point transformations provides us with twofamilies of power-law potentials for the Brans-Dicke field, in which the powersare functions of the Brans-Dicke parameter $\omega_{BD}$ and the parameter $w$.The existence of the Lie symmetry in the Wheeler-DeWitt equation is equivalentto the existence of a conserved quantity in field equations and withoscillatory terms in the wavefunction of the universe. This enables us to solvethe field equations. For a specific value of the conserved quantity, we find aclosed-form solution for the Hubble factor, which is equivalent to acosmological model in general relativity containing two perfect fluids. Thisprovides us with different models for specific values of the parameters$\omega_{BD},$ and $w$. Finally, the results hold for the specific case wherethe Brans-Dicke parameter $\omega_{BD}$ is zero, that is, for the O'Hanlonmassive dilaton theory, and consequently for $f\left( R\right) $ gravity in themetric formalism.

Classical and Quantum Solutions in Brans-Dicke Cosmology with a Perfect Fluid