I have been reading about dynamical systems, flows and invariant sets for a while. I have a question that if we are given a system of differential equations, which can be written as
$$\dfrac{d \bar{x}}{dt} = \bar{F} \left( \bar{x}, t \right)$$
can we actually find the flow or the invariant sets without solving the system? In particular, I would like to know the flow/invariant sets of the system of differential equations
\begin{align} x_1' &= x_2 \\ x_2' &= a \left( t \right) g \left( x_1 \right) + \dfrac{b \left( t \right)}{x_1^{\alpha}} + p \left( t \right) \end{align}
where the only information available with us is that the functions $a, b, p$ are periodic with period $T$ and are $L^1$. The function $g$ can be anything and $\alpha > 0$. The system is defined for $x_1 > 0$ and $x_2 \in \mathbb{R}$.
I do understand what we mean by invariant sets and flow of this equation. But without the knowledge of the solutions, which cannot be found in the general case, I cannot really proceed with finding the flow and the invariant sets. Any help will be appreciated!