Imagine a body with mass m moves through a pipe through the centre of earth. The gravitational force is given by $\vec{F}=-C \vec{r}$ where $C$ is constant. I want to determine the 1D equation of motion of the body and solve it for the case that the body starts at the surface of the earth with initial velocity $v_0=0$.

My attempt: $F=m\ddot{r}=-Cr$ so $r=Ae^{x\sqrt{-C/m}}+Be^{-x\sqrt{-C/m}}$. But I think this isn't right, shouldn't the body stay at the centre of the eart? Because I think the attraction is largest there.