What is an example of a balanced diagnostic equation, and how does it work?

A diagnostic equation is one in which there is no time derivative. This may mean one of the two: either all the variables used in the equation are independent of time, or all the variables are evaluated at the same point in time (at the same time instant). Some examples of diagnostic equations are ideal gas law, hydrostatic equation, etc.
The hydrostatic equation is given as:
dp/dz = -rhog
where p is the pressure, rho is density, g is acceleration due to gravity and z is the geometric height. In this equation, there is no time variable. And this equation describes the balance between gravitation force (directed downwards) and pressure gradient force (directed upwards). And hence, this is an example of a balanced diagnostic equation.
Similarly, the ideal gas law is given as:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant and T is the temperature. This equation does not have any time variable or time derivative. It is understood that while all these variables (P, V, n, and T) may have different values at different time points, the equation is applied at a given time instant where the measured values of these parameters are known. In other words, the equation is valid at all time points and that any change in the value of any of these variables will cause a change in the values of one or more variables. Hence this is another balanced diagnostic equation.
Another class of equations is the prognostic equations. These have time derivative of a quantity and hence can be used to determine the value of that quantity at any other time instant (given the values of other parameters at that time). The continuity equation is a good example of a prognostic equation.
Hope this helps.