this scalar equation. Having obtained a solution of (\[2nd\]), one can integrate the gravitational equations in the bulk using, for example, Gaussian normal coordinates [@SMS; @Shtanov].

The brane filled by a homogeneous and isotropic ideal fluid is described by the Friedmann–Robertson–Walker (FRW) metric $$ds^2=-dt^2+a^2(t)\left[\frac{dr^2}{1-\kappa r^2}+r^2d\Omega^2\right]\, ,$$ where $\kappa=-1$, $0$, or $+1$. The stress–energy tensor of the ideal fluid takes the form $$\tau_{ab}=(\rho+p)u_{a}u_{b}+ph_{ab} \, ,$$ where $\rho$ is the energy density, and $p$ is the pressure. In this paper, we assume a simple equation of state $p = w \rho$, where $w$ is constant.

Equation (\[2nd\]) leads to the following cosmological equation in the braneworld theory: $$\label{second}
9M^{6}\left[\frac{\ddot{a}}{a}+H^2+\frac{\kappa}{a^2}-\frac{\Lambda_{\rm
b}}3\right] +\left[\rho(1+3w)-2\sigma+6m^2\frac{\ddot{a}}{a}\right]
\left[\rho+\sigma - 3m^2\left(H^2+\frac{\kappa}{a^2}\right)\right]=0\, ,$$ where $H \equiv \dot a / a$ is the Hubble parameter. After multiplying this equation by $a^3\dot{a}$ and using the conservation of energy $$\dot{\rho}+3H(\rho+p)=0 \, ,$$ one can integrate Eq. (\[second\]) and obtain the first-order differential equation [@Shtanov; @SS] $$\label{1st}
m^4\left(H^2+\frac{\kappa}{a^2}-\frac{\rho+\sigma}{3m^2}\right)^2=
M^{6}\left(H^2+\frac{\kappa}{a^2}-\frac{\Lambda_{\rm
b}}{6}-\frac{C}{a^4}\right)\, ,$$ where $C$ is the integration constant corresponding to the black-hole mass of the Schwarzschild–(anti)-de Sitter solution in the bulk and associated with what is called “dark radiation.”

Equation (\[1st\]) can be solved with respect to the Hubble parameter: $$\label{first}
H^2+\frac{\kappa}{a^2}=\frac{\rho+\sigma}{3m^2}+\frac2{\ell^2}
\left[1\pm\sqrt{1+\ell^2\left(\frac{\rho+\sigma}{3m^2}-\frac{\Lambda_{\rm
b}}{6} -\frac{C}{a^4}\right)}\right]\, 