We proceed to obtain an exact analytical solution of the Brans–Dicke (BD) equations for the spatially flat ($k=0$) Friedmann–Lamaître–Robertson–Walker (FLRW) cosmological 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, $\lambda$-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 solutions. 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 cosmological model, we find the Lie point symmetries, $\lambda$-symmetries, and DPs, and obtain the basic quantities of the extended PS method (which are the null forms and the integrating 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 solving 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.

我们继续获得空间平坦 ( BD) 方程的精确解析解 ($k=0$）弗里德曼-拉迈特-罗伯逊-沃克（FLRW）宇宙学模型在宇宙学常数不存在和存在的情况下。我们用来求解BD方程场方程的求解方法称为“对称群不变量法”（ISG法）。该方法基于扩展的 Prelle-Singer (PS) 方法，并采用李点对称性，$\lambda$-对称性和达布多项式（DP）。事实上，ISG 方法试图提供与保持常微分方程 (ODE) 不变的单参数李变换组相关的两个独立的一阶不变量作为解。应该注意的是，对于可积 ODE，ISG 方法保证了这两个不变量的提取。在这项工作中，对于 FLRW 宇宙学模型中的 BD 方程，我们找到了 Lie 点对称性，$\lambda$-对称性和DP，并获得扩展PS方法的基本量（即零形式和积分因子）。通过使用扩展的 PS 方法，我们找到了两个独立的一阶不变量，通过将这些不变量作为代数方程组求解，可以同时获得适当的宇宙学解。这些解决方案很丰富，因为它们包括许多已知的特殊解决方案，例如 O'Hanlon–Tupper 真空解决方案、Nariai 解决方案、Brans–Dicke 粉尘解决方案、膨胀解决方案等。