I would suggest using Lagrangian mechanics to solve the problem. Basically you chose a convenient coordinate system (lattitude, longitude, and height? or perhaps Euler angles?), and then you write the Lagrangian in that coordinate system as a function of your chosing variables, and their time derivatives.
For this simple problem, the Lagrangian L of the missile will be the kinetic energy T in an earth-centered inertial frame minus the potential enregy V in an ECI frame.
Then you use Lagrange's equations to get the equations of motion for the missile.
There's an overview at the Wikipedia
http://en.wikipedia.org/wiki/Lagrangian_mechanics
it may not be clear enough if you are not familiar with the subject. You may have to consult a textbook if you want a really detailed explanation. The quick overview is that you have a function L, called the Lagrangian which is written in the form
L(x, x', t), where x is is a coordinate, x' is it's time derivative, and t is time.
Then Lagrange's equations give you the equations of motion directly from the Lagrangian
<br />
\frac{d}{dt}\left(\frac{\partial L}{\partial x'}\right) =\frac{\partial L}{\partial x}<br />
A simple example - in cartesian coordinates in a potential V with only one coordinate x
L(x,x') = .5*m*x'^2 - V(x)
(note that this is kinetic energy minus potential energy).
Then
d/dt(m*x') = -\partial V/\partial x
For systems with more than one coordinate, there is one Lagrange's equation for each independent coordiante (variable).