I am not asking for a solution of the following problem that appears in my assignment. However, I don't understand the question and I would like someone to explain the what the question actually is asking AND also hints for a solution as in where to look for or what to think in solving this,
Express the geodesic equation as a differential equation for $x^{\mu}(\tau) .$ What is the most general transformation of the Christoffel symbols that will leave these equations invariant?
(The question assumes metric compatibility and that the connection torsionfree.) The Christoffel can change if there's a transformation $x\rightarrow\bar{x}$ and $g_{\mu\nu}\rightarrow\bar{g}_{\mu\nu}$. Also it can change if $\tau\rightarrow{\tau^\prime}$, via, $$\frac{d^2 x^\mu}{d\tau^2}+\Gamma^\mu_{\alpha\beta}(x(\tau))\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}=0\tag{1}$$ What transformations are the question asking for (may be both! Or maybe a different kind of transformation for Christoffel symbols altogeher)? How to approach the question?