$(d g/d\psi)d\psi \rightarrow d\tilde{\psi}$, the line element in $5D$ can be written as $$\label{5D line element for the barotropic case}
dS^2 = \frac{D d\tilde{t}^2}{\tilde{H}^{(3n + 1)c}} - A \tilde{H}^{2 a \alpha } dx^2 - B \tilde{H}^{2 a \beta } dy^2 - C \tilde{H}^{2 a \gamma } dz^2 - \frac{D  d\tilde{\psi}^2}{\tilde{H}^{(3n + 1)c}},$$ where $\tilde{H} \equiv \left(\tilde{t} + E \tilde{\psi}\right)^{\frac{1}{a + b + (3n + 1)c}}$; $E$ is an arbitrary constant for $n = 1/3$, but $E = \pm 1$ for any other $n \neq 1/3$; $D$ is a positive constant introduced for dimensional considerations.

### Kasner universe in $5D$

We immediately note that the case where $E = 0$, which requires $n = 1/3$, gives back the well-known Kasner universe in $5D$. In fact, setting $\tilde{t} \propto \tau^{(a + b + 2c)/(a + b + c)}$ the line element (\[5D line element for the barotropic case\]) reduces to $$\label{Kasner solution in 5D}
dS^2 = d\tau^2 - A \tau^{2q_{1}} dx^2 - B \tau^{2 q_{2}}dy^2 - C \tau^{2 q_{3}}dz^2 \pm D \tau^{2 q_{4}}d\psi^2,$$ where $$\label{defifition of the q's}
q_{1} = \frac{\alpha a}{(a + b + c)},  \;\;\;q_{2} = \frac{\beta a}{(a + b + c)}, \;\;\;q_{3} = \frac{\gamma a}{(a + b + c)}, 