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Abstract
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We proceed to obtain an exact analytical solution of the Brans–Dicke (BD) equations for
the spatially flat (k = 0) Friedmann–Lamaˆıtre–Robertson–Walker (FLRW) cosmologi-
cal model in both cases of the absence and presence of the cosmological constant. The
solution method that we use to solve the field equations of the BD equations is called
the “invariants of symmetry groups method” (ISG method). This method is based on
the extended Prelle–Singer (PS) method and it employs the Lie point symmetries, λ-
symmetries, and Darboux polynomials (DPs). Indeed, the ISG method tries to provide
two independent first-order invariants associated to the one-parameter Lie groups of
transformations keeping the ordinary differential equations (ODEs) invariant, as solu-
tions. It should be noted for integrable ODEs that the ISG method guarantees the
extraction of these two invariants. In this work, for the BD equations in FLRW cos-
mological model, we find the Lie point symmetries, λ-symmetries, and DPs, and obtain
the basic quantities of the extended PS method (which are the null forms and the inte-
grating factors). By making use of the extended PS method we find two independent
first-order invariants in such a way that appropriate cosmological solutions from solv-
ing these invariants as a system of algebraic equations are simultaneously obtained.
These solutions are wealthy in that they include many known special solutions, such as
the O’Hanlon–Tupper vacuum solutions, Nariai’s solutions, Brans–Dicke dust solutions,
inflationary solutions, etc.
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