Undergrad Variable mass F=ma versus Rocket F=ma

[walking]

In the book by Tipler & Mosca, the section on F=ma for variable mass derives the following equation:

$\mathbf{F}_{ext}+\frac{dM}{dt} \mathbf{v}_{rel}=M\frac{d\mathbf{v}}{dt}$

where $\mathbf{F}_ext$ is the external force on the system as a whole (ie not just the variable mass sub-system, but the system as a whole), $\frac{dM}{dt}$ is the rate at which the mass of the variable mass sub-system is changing, $\mathbf{v}_{rel}$ is the velocity of the changing mass (whether incoming or outgoing) relative to the variable mass subsystem, $M$ is the mass of the variable mass subsystem at time t, and finally, $\mathbf{v}$ is the velocity of the variable mass subsystem at time t.

Now I may have misunderstood this equation, but if not then I am slightly confused as to the authors's subsequent derivation of the rocket equation. When deriving this latter equation, they seem to interpret the general variable mass equation completely differently, and $F_{Ext}$ no longer represents the total external force on the system for example. Here is their rocket equation:

$M\mathbf{g}-R\mathbf{u}_{ex}=M\frac{d\mathbf{v}}{dt}$,

where $M$ is the rocket's mass at time t, -R is the rate at which fuel-mass is leaving, $\mathbf{u}_{ex}$ is the velocity of the leaving fuel relative to the rocket.

As I said, the authors seem to have changed their interpretation of $F_{ext}$, because when I use the original variable mass equation on a rocket, I get a different answer. Here is my derivation of the rocket equation based on my understanding (a possibly erroneous one) of the general variable mass equation.

For a rocket-fuel system, $F_{ext}$ in the general equation should be the total gravitational force on the rocket and the fuel. If we let the initial mass of the rocket-fuel be $M_0$, then $F_{ext}=M_0\mathbf{g}$. Then deriving the rest of the rocket equation as the authors do, I get

$M_0\mathbf{g}-R\mathbf{u}_{ex}=M\frac{d\mathbf{v}}{dt}$,

which doesn't agree with their equation. The problem, as I highlighted above, seems to be the authors's use of $F_{ext}$. Here are their derivations of both equations.

General variable mass F=ma:

Rocket equation:

 

[DrStupid]

When deriving this latter equation, they seem to interpret the general variable mass equation completely differently, and $F_{Ext}$ no longer represents the total external force on the system for example.

Maybe they just use a little trick: For the current acceleration of the rocket it doesn't matter weather it is already burning for hours or if it has just been started (because exhausted reaction mass doesn't interact with the rocket anymore). Therefore you can always assume M = Mo and neglect the external force acting on the exhaust.

Edit: After reading the sourece again I'm quite sure this is the case and it is not even a trick. The "whole system" that the external force is acting on consists of the rocket (with the mass M) and the reaction mass that is currently exhausted (with the mass $\Delta M$). Reaction mass that has been exhausted in the past (with the mass $M_0 - M - \Delta M$) is not included.