I'm trying to do some exercises of my book to better understand the concept of parallel transport.
Excercise: Consider the affine connection on $\mathbb{R^2}$ defined by the Christoffel symbols $\Gamma^i_ {j,k} = 2 $ if $i=j=k=2 $, $\Gamma^i_ {j,k} = 0$ otherwise and the curve:
$\sigma: \mathbb{ R} \rightarrow \mathbb{ R^2}$
$ t \rightarrow (e^{-4t}+5, 3t+7) $
Find the vector field given by the parallel transport along the curve $\sigma$ of its tangent vector $\sigma(2)$.
There are no numeric examples on my book and I can't understand what should I do to solve this excercise. It's marked as an easy exercise, but I'm really lost.
Edit: my problem is that I don't understand how to include in that ODE $\sigma (2)$. I have computed that $y^1 = const$ and $y^2 = e^{6t}/6 + const$. But I don't know what to do next.
Hint: In your book you probably have the ODE for the parallel transported vector $y(t)$: $$ \dot{y}^i + \Gamma^i_{jk} \dot{\sigma}^j y^k = 0 . $$ This takes a particular simple form in your example.