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The premise here is the "continuous Petri net" approximation, which can be pictured as having buckets filled with fluid at the places; the transitions are like pumps that are draining their input buckets and supplying their output buckets.
Let's start with the simplest case of a Petri net, which has just one species $U$, and one transition $Z$. (I guess you could have a net with zero species that is completely vacant.) Suppose that $Z$ has $m$ input connections to $U$, and $n$ output connections to $U$. Let $\alpha$ be the rate constant for $Z$.
Let $u(t)$ be the continuous amount stored at $U$ at time $t$.
Then the firing rate for $Z$ at time $t$ is $\alpha u(t)^m$.
The general, time-invariant "direction vector" for $Z$ is $(n - m)$ — this is how many tokens would get added for each firing of the transition, back in the discrete model. Let's call this $DirVec(Z)$.
Then the rate equation states that:
$$u'(t) = rate(u(t)) \cdot DirVec(Z) = \alpha \cdot u(t)^m \cdot (n - m)$$ So the general form of the equation that is raised is:
$u'(t) = \beta \, u(t)^m$
When $m = 0$, the general solution is $u(t) =$ affine function of $t$.
When $m = 1$, the general solution is $u(t) =$ exponential function of $t$.
When $m = 2$, a solution is $u(t) = - 1 /(t + const)$. I found this one on the web. Is this the general form of the solution for $m = 2$?
I find this to be a curious sequence of functions: linear, exponential, reciprocal of a polynomial.
Does anyone know a general solution, that covers all cases of $m$?
See, here is the pull of the formula-based approach.