Dynamics of a Rocket I am interested in modelling the trajectory of a rocket from the Earth to the Moon by solving a differential equation numerically. Below are some key facts and assumptions I am using. I want to make sure that I have not made any serious mistakes, nor disregarded any necessary facts.
We will consider the following equation,
$$
\vec{T} + \vec{c}(\vec{r})\dot{\vec{x}} + \vec{G}(\vec{r})= m(t) \ddot{\vec{x}},
$$
where $T$ is the constant rocket thrust, $c$ denotes air resistance and is a function of radial distance from the earth, and the rocket has mass that drops at a rate that is constant with respect to time (we are assuming that a constant amount of fuel is always used for constant rocket thrust -- is this a valid assumption?).
Now a question:


*

*The trajectory of the rocket is not straight; how do we incorporate parabolic motion into the numerics?

 A: Your reason for asking about parabolic motion is unclear (rockets generally don't travel on parabolic trajectories, especially not when under thrust or in an atmosphere), but you seem to be coming at things the wrong way, trying to directly implement the results of gravitational forces instead of just integrating them.
Since you're already doing a numerical solution, you can't use an analytical one for the gravitational part of the problem...you have other forces influencing your trajectory. However, since you're already numerically computing the motion of the rocket due to the forces on it, it should be straightforward to model drag and gravity as just two different forces on the rocket.
A: The trajectory is in 3D. Your equation contains some vectors, $\vec{r}$ and $\vec{x}$. It is not clear what these vectors are, or why the air drag coefficient is a vector, which depends on the vector $\vec{r}$. If $\vec{G}$ is the gravitational force, then it doesn't depend on a vector. And does that include both the attraction due to the Earth and to the Moon?
The rocket won't have a parabolic trajectory, but it will have a curved trajectory. You need to make sure you define your coordinate system correctly (is it attached to the Earth? is it an inertial frame?), and write the separate components of the equation of motion. 
A: You seem to have the right idea with your equation, but some practical difficulties. First, I would recommend combining your independent variables into simply $\vec{r}$ rather than the $\vec{x}$ and $\vec{r}$, as you have it right now. Rocket motion is very complicated, however, and consider that nearly all of your forces will be functions of $\vec{r}$ and $\dot{\vec{r}}$ (e.g. air resistance is a function of velocity as well as position - you currently have it as only a function of position). After solving the final equation (presumably numerically, since a symbolic result for such a complex relationship will not be closed-form), you will get a function which models the motion of your rocket at a given time.
To account for "parabolic motion," by which I think you mean "turning," I would make thrust a function of time. For example, you may want to fire the thrusters 60 minutes into the flight to adjust the orbit, so you would define $T$ to have a certain magnitude and direction when $t = 60 \text{ min}$.
