To derive the geodesics for any metric, write out and solve the Euler-lagrange equations for the Lagrangian F = gμν(dxμ/ds)(dxν/ds).
For Schwarzschild this is (in the equatorial plane) F = (1-2m/r)-1(dr/ds)2 + r2(dθ/ds)2 - (1-2m/r)(dt/ds)2 . There are three first integrals: one for the t coordinate, one for the θ coordinate, and F itself. The result can be expressed as a differential equation for r(θ).
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