Suppose that a particle of charge $q$ is exactly one meter from an oppositely charged particle (same magnitude of charge). The initial velocity of this particle is $0$. From Coloumb’s law and Newton’s second law, the initial value problem will be the following:
$r”(t) = -\frac{kq^{2}}{m}\cdot \frac{1}{(r(t))^{2}}$, $r(0)=1$, $r’(0)=0$, where $m$ is the Mass of the particle, and $r(t)$ is the distance between the particles $t$ seconds after release. Does there exists closed form solution to this IVP? If $r’(0) \neq 0$, a closed form solution can be found by writing $r(t) = A (C\pm t)^{\frac{2}{3}}$. But if $r’(0)=0$, then $C$ must be zero, and $r(0)$ also will have to be zero, contrary to the initial condition. Does another closed form solution exist? Or, even if a closed form solution does not exist, can it be written in terms of a series? I am unsure how to rewrite $\frac{1}{(\sum_{n = 1} a_{n}x^{n})^{2}}$ as a series. Why or why not?