How to calculate a Satellite position at a given time

I am trying to write a program to be able to "model" a Low Earth Orbit satellite's position at for a given time in the future. However I do not have a TLE for the desired satellite, I simply have a state-vector (Epoch and XYZ Position and Velocity). I am aware that this may not be a accurate projection for any significant amount of time, but this is all I have to work with.

What would the recommended steps be for working this out? Could I convert the state-vector into orbital elements somehow and then use those elements to calculate the position in respect to time? I am rather new to orbital mechanics so I apologize if this is a simple question. I have done some research and it appears the the orbital elements route seems like the route that most others have taken. However once I get those orbital elements, how do I calculate the satellites position?

Once I have an algorithm worked out I have other satellites with TLE's I plan to decompose into state-vectors and then compare the calculated state vector position with the calculated TLE position to ensure I am in the right ballpark.

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• You simply don't have enough data to start with. You need velocity / orbital period and then you might stand a chance of getting something. Unless you have many values for Epoch and XYZ. – MrPhooky May 11 '17 at 12:10
• @MrPhooky I disagree. Knowing the initial state and then finding the Earth state should be sufficient to setup and solve the differential equation, assuming no other gravitational forces. – barrycarter May 11 '17 at 17:15
• @barrycarter That would be my mistake, I did not see that his state included the velocity as I would have expected. Apologies. – MrPhooky May 16 '17 at 10:05