My question is: Is it possible to (numerically) solve the Friedmann Equations, starting from some epoch if we do not explicitly know the equation of state $w$?
Let us suppose we have a component, which at an initial time $t_i$ (when the scale factor is $a_i$) has a density $\rho(a_i)$ and pressure $P(a_i)$. I am interested in self-consistently studying the evolution of $\rho(a)$ and $P(a)$ (and hence $w(a)$) as the univese expands.
The way I was proceeding was as follows. First, we take the equations,
$$H^2(t)=\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3}\rho,\quad\quad\quad\quad \dot{H}(t)+H^2(t)=\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\big(\rho+3P\big)$$
from which we can get the continuity equation,
$$\dot{\rho}+\frac{3\dot{a}}{a}\big(\rho+P\big)=0$$
Also, the second Friedmann equation can be modified with the help of the first to get,
$$\dot{H}(t)+4\pi G\big(\rho+P\big)=0$$
In the next step, I can convert all $t$ dependencies to $a$ dependencies (primes denote derivatives wrt $a$, $\frac{d}{dt}=aH\frac{d}{da}$),
$$\rho'+\frac{3}{a}\big(\rho+P\big)=0, \quad\quad\quad\quad aHH'+4\pi G\big(\rho+P\big)=0$$
The question is:
Is it possible to solve these equations in a self-consistent manner, or is there too little information? Am I missing something simple, due to which I do not get an expression for $P'$?