I'm working using the standard FRW metric,
$$ds^2=dt^2-a^2\left [\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta d\phi^2)\right ]$$
Using the definition of the Christoffel symbols,
$$\Gamma^c_{ab}=\frac{1}{2}g^{cd}(g_{ad,c}+g_{bd,a}-g_{ab,d})$$
I've found the non-zero Christoffel symbols for the FRW metric, using the notation $(t,r,\theta,\phi)=(0,1,2,3)$,
$$\Gamma^{0}_{11}=\frac{a\dot{a}}{1-kr^2} \ \ ,\ \ \Gamma^{0}_{22}=a\dot{a}r^2 \ \ ,\ \ \Gamma^{0}_{33}=a\dot{a}r^2 \sin^2\theta \ \ , \ \ \Gamma^{1}_{11}=\frac{kr}{(1-kr^2)}$$
$$\Gamma^{1}_{01}=\Gamma^{1}_{02}=\Gamma^{1}_{03}=\frac{\dot{a}}{a} \ \ ,\ \ \Gamma^{1}_{22}=-r(1-kr^2) \ \ ,\ \ \Gamma^{1}_{33}=-r(1-kr^2) \sin^2\theta$$
$$\Gamma^{2}_{12}=\Gamma^{3}_{13}=\frac{1}{r} \ \ ,\ \ \Gamma^{2}_{33}=-\sin\theta \cos\theta \ \ ,\ \ \Gamma^{3}_{23}=\frac{\cos\theta}{\sin\theta}$$
Now I'm trying to derive the geodesic equations for this metric, which are given as,
$$\frac{d^2x^\mu}{ds^2}+\Gamma^{\mu}_{\nu\lambda}\frac{dx^\nu}{ds}\frac{dx^\lambda}{ds}=0$$
For example, for $\mu=0$, I get that,
$$\frac{d^2t}{ds^2}+\frac{a\dot{a}}{1-kr^2}\left (\frac{dr}{ds}\right )^2+a\dot{a}r^2\left (\frac{d\theta}{ds}\right )^2+a\dot{a}r^2 \sin^2\theta\left (\frac{d\phi}{ds}\right )^2=0$$
However, when I checked with this document in the section of geodesics (http://popia.ft.uam.es/Cosmology/files/02FriedmannModels.pdf), they get,
$$\frac{|u|}{c}|\dot{u}|+\frac{\dot{a}}{a}|u|^2=0$$
where $\frac{du^0}{ds}=\frac{|u|}{c}|\dot{u}|$ and $u^\mu=\frac{dx^\mu}{dt}$.
I'm not sure what I'm doing wrong or if it's just a matter of convention. I also checked this question (Geodesics for FRW metric using variational principle) but the FRW metric is slightly different, so it didn't help.