Solving Differential Equations in General Relativity

[Jianbing_Shao]

TL;DR

differentail equations, parallel transport equations

In genaral relativity, how to solve differential equations is seldom be discussed. I want to know how to sole the differential equations like this:
$$\partial_kv^i(x)+\Gamma^i_{jk}(x)v^j(x)=\partial_kA^i(x)$$
Here $\Gamma^i_{jk}(x)$ is connection field on a manifod and $A^i(x)$ is a vector field on the manifold. then how to get $v^i(x)$, Perhapa we can not find a global vector field which can satisfy the differential equations, But we can still find a solution on a partilular curve. then how to find it?

 

[PeterDonis]

In genaral relativity, how to solve differential equations is seldom be discussed.

That's because it's a general mathematical subject, not specific to general relativity. So it's discussed under the general math subject of solving differential equations.

I want to know how to sole the differential equations like this

This question is much, much too general for a forum discussion. You need to study a math textbook on solving differential equations. If there is a particular case that arises in a particular problem in general relativity and you're having trouble solving that particular case, you can open a new thread asking about that particular case (and showing your attempts at solving it and where you're getting stuck). But this thread is closed since giving classes in general math techniques is outside the scope of PF.