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When the force ##F## and its resulting acceleration ##a## have the most general form, the acceleration can depend on the position ##x##, time ##t## and speed ##v##. Newton's second law is given by ## \frac {d^2x}{d^2t}= a(x,t,v)##.

When the acceleration is only a function of position, i.e. ##a(x)##, we can use the chain rule ## \frac {dv}{dt} = \frac {dv} {dx} v ## and Newton's 2nd law becomes $$ v dv = a(x) dx $$ and we can solve for ##v(x)##.

But what if the acceleration is instead a function of both time ##t## and position ##x##, i.e. ## \frac {d^2x}{d^2t}= a(x,t)##? What kind of approach do we use? What kind of chain rule can we use? How do we solve for the speed ##v##?

Thanks!