I was reading the deduction of the geodesic eq. and at the end the expression for acceleration was expressed by 
\begin{equation}
 \ddot{x}^{m}=-\Gamma\_{ij}^{m}\dot{x}^{j}\dot{x}^{i}
\end{equation}
In another chapter that explains the equations of motion for a particle it reads 
\begin{equation}
 a^i=\frac{\delta}{\delta t}v^i =\frac{d}{dt}v^i+\Gamma\_{jk}^i v^jv^k =\frac{d^2}{dt^2}x^i+\Gamma\_{jk}^i\frac{d}{dt}x^j\frac{d}{dt}x^k
\end{equation}
Shouldn't both agree? I can't conciliate both as true becase one has the same term but as negative. Does the first one only works for geodesic paths? What considerations am I missing?

I need to resolve this conflict because later one I want to have the expression for the Jolt/jerk which reads 
\begin{equation}
 \dddot{q}\,^k=-\left(\frac{\partial}{\partial q^\gamma}\Gamma\_{\alpha\beta}^k-2\Gamma\_{m\beta}^k\Gamma\_{\alpha\gamma}^m\right)\dot{q}^\alpha\dot{q}^\beta\dot{q}^\gamma
\end{equation}

which am also having trouble to derive. I could only develop until the following:  

Since 
 \begin{equation}
 a^i=\frac{\delta}{\delta t}v^i =\frac{d}{dt}v^i+\Gamma\_{jk}^i v^jv^k =\frac{d^2}{dt^2}x^i+\Gamma\_{jk}^i\frac{d}{dt}x^j\frac{d}{dt}x^k
\end{equation}
I differentiate it to have 
\begin{equation}
\begin{split}
J^i&=\frac{d}{dt}a^i+v^l\Gamma\_{lm}^i a^m 
\\
&=\frac{d}{dt}\left[\frac{d}{dt}v^i+\Gamma\_{jk}^i v^jv^k \right]+v^l\Gamma\_{lm}^i \left[\frac{d}{dt}v^m+\Gamma\_{jk}^m v^jv^k \right]
\\
&=\frac{d^2}{dt^2}v^i+\frac{d}{dt}\left[\Gamma\_{jk}^i v^jv^k \right]+v^l\Gamma\_{lm}^i \frac{d}{dt}v^m + v^l\Gamma\_{lm}^iv^k\Gamma\_{jk}^mv^j 
\end{split}
\end{equation}
which is not completely the same equation but I fell it's getting there but I believe both of my questions are intimately related.

Thanks in advance!