I was solving a physics problem and I was solving it using a reference particle. During the solving , I came across this differential equation.
$$dx= \frac{GM}{x^2}(dt)^2$$ G,M are constants.
I haven't ever integrated such an equation. Is this integral solvable?
You probably mean motion in a gravity field, $$ \frac{d^2x}{dt^2}=-\frac{GM}{|x|^2}\hat x=-\frac{GMx}{|x|^3}. $$ There are two ways to integrate once: Multiply with $2\dot x$ and get $$ \frac{d}{dt}|\dot x|^2=\frac{GM}{|x|}+C $$ and the cross product with $x$ to get $$ \frac{d}{dt}(\dot x\times x)=\ddot x\times x=0\implies \dot x\times x = D. $$ These lead to the Kepler laws of planetary motion.