I am a first year physics student. I am trying to figure out how to compute position in terms of time for an object falling through non uniform gravity towards the earth, and by extension towards any body. There are two formulas that can help with this: $$-{GM_E \over r^2}={d^2 r \over dt^2} $$ which is a second order differential equation. A presumably simpler approach would be to use the energy formula for velocity, that is $$K = {GM_{E}m} {\left({1 \over r} -{1\over r_i} \right)}={1 \over2}mv^2 $$ Solving for velocity would yield the first order differential equation : $$v={dr\over dt}=\sqrt{{2GM_{E}} {\left({1 \over r} -{1\over r_i} \right)}} $$ Apparently, however, there is no elementary function that can be used as an anti derivative for this function. I searched the internet far and wide, but I could not find any straightforward answer to this.Obviously the answer is far from straightforward, otherwise it would be taught in physics texts. However, it appears to be far too fundamental a concept to not have a known solution for.
P.S. If the answer involves anything such as Lagrange multipliers and the like, if whoever answers this question doesn't mind, please provide at least a basic explanation of it, because I haven't yet taken anything beyond second year calculus. Thank you.