finite at small $t$ as given in Fig. \[Figg2\].

Particular Solution {#part2}
===================

Another particular solution for Eq. (\[init\]) can be obtained, when assuming that the dependence of $u$ on $H$ can be given by the polynomial in Eq. (\[init-polyn\]) $$\label{init-partc2}
u=b_{1}H^{2}+b_{2}H,$$ where $b_{1}$ and $b_{2}$ are constants. Some simple calculations show that this form is a solution of the initial equation, Eq. (\[init2\]), if $$\begin{aligned}
b_1 &=&-\frac{3}{2}\frac{1+\gamma }{1-r}, \\
b_2 &=& \frac{1}{r\alpha }.\end{aligned}$$ $b_2$ is identical to $B$ in Eq. (\[paramsab\]). $r$ and $\gamma$ have to satisfy the compatibility relation $$r=\frac{2-\gamma}{2+\gamma^2}.$$ Integrating Eq. (\[init-partc2\]) results in $$\begin{aligned}
 \label{Eq:Ht}
H(t) &=& \frac{b_2\exp(-b_2 t)}{1-b_1\exp(-b_2 t)},\end{aligned}$$ where minus sign in the exponential function refers to flipping the integral limits. This was not necessary while deriving the expressions given in Section \[approx\]. The free parameter [P]{} compensates it. The geometric and thermodynamic quantities of the Universe read $$\begin{aligned}
a(t)&=&a_0\left(\frac{\exp(b_2t)-b_1}{\exp(b_2t)}\right)^{1/b_1}, \label{partc2a} \\
\rho(t)&=& 3 \left(\frac{b_2\exp(-b_2 t)}{1-b_1\exp(-b_2 t)}\right)^2, \label{partc2rho}\\
T(t)&=& 3^r\;\beta \left(\frac{b_2\exp(-b_2 t)}{1-b_1\exp(-b_2 t)}\right)^{2r}, \label{partc2T}\\
\Pi(t)&=& \frac{b_2^2 \left[2\exp(b_2t)-3\gamma\right]}{\left[\exp(b_2t)-b_1\right]^2}, \label{partc2Pi}\\
q(t) &=& \exp(b_2t)-1. \label{partc2q}\end{aligned}$$ Obviously , we notice that the scale parameter in Eq. (\[partc2a\]) looks like Eq. (\[approx-a\]), which strongly depends on the free parameter ${\cal
  P}$. The other geometric and thermodynamic quantities find similarities 