When is a system of equations 'closed'? In Sean M Carroll's Introduction to General Relativity: Spacetime and Geometry, after deriving the Tolman-Oppenheimer-Volkoff equation on page 233, Chapter 5, he says:
"To get a closed system of equations, we need one more relation: the equation of state."
What does 'closed system of equations' mean? Does it mean that the differential equation has a closed form solution? In that case, how does that happen upon knowing the equation of state?
 A: "Closed system of equations" means the number of unknown variables is equal to the number of equations supplied with boundary and initial conditions sufficient to unambiguously find the searched variables.
In case of "the equation of state", it is often a solution (an expression rather than a differential equation) for some variable. In othe words, for this variable the approximate solution is considered good in a given context.
A: In this context, a system of equations is said to be closed when it constrains fully the dynamics of the system. Heuristically, it means that there are as many equations as there are independent degrees of freedom.
For example, the Maxwell equations :
$$\begin{array}{ccc} 
\nabla \cdot E = \frac{\rho}{\epsilon_0} && \nabla \times E = -\partial_t B\\
\nabla\cdot B = 0 && \nabla\times B = \mu_0 j + \mu_0\epsilon_0\partial_tE
\end{array}$$
are a closed set of equations if the source $\rho$ and $j$ are considered as a fixed background. If they are considered as independent variables, you would need to add equations to describe how the dynamics of $\rho$ and $j$ is determined by the electromagnetic field.
