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(d

^{2}x

^{m}/ds

^{2}) + Γ

^{m}

_{ab}(dx

^{a}/ds)(dx

^{b}/ds) = 0

In the case of the Godel metric, the geodesic equations that I was able to derive after deriving the Christoffel symbols are as follows:

(d

^{2}x

^{0}/ds

^{2}) + 2(dx

^{0}/ds)(dx

^{1}/ds) + e

^{x}(dx

^{1}/ds)(dx

^{3}/ds) = 0

(d

^{2}x

^{1}/ds

^{2}) + e

^{x}(dx

^{0}/ds)(dx

^{3}/ds) + ( e

^{2x}/ 2 )(dx

^{3}/ds)(dx

^{3}/ds) = 0

(d

^{2}x

^{2}/ds

^{2}) = 0 (This one is easy to solve. It is just a straight line x

^{2}(s) = As + B where A and B are constants).

(d

^{2}x

^{3}/ds

^{2}) - (2 / e

^{x})(dx

^{0}/ds)(dx

^{1}/ds) = 0

Now can anyone either direct me to some free or cheap software that I could use to solve these equations, or give me a method that would commonly be used to solve these?

Thank you.