This might seem like an easy question, I would appreciate any help you can provide.
For the force F can be factored as below, is the equation of motion (EOM) integrable?
a) $F\left({x}_{i},t\right)=f\left({x}_{i}\right)g\left(t\right)$
b) $F\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right),t\right)=f\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)g\left(t\right)$
c) $F\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right),{x}_{i}\right)=f\left({x}_{i}\right)g\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)$
My attempt:
a) EOM not integrable (why though? how can I be sure?)
b) $m\displaystyle\frac{d}{dt}\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)=f\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)g\left(t\right)=>m\displaystyle\int{\dfrac{1}{f\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)}}d\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)=\displaystyle\int{g\left(t\right)}dt$
c) $m\displaystyle\frac{d}{dt}\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)=g\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)f\left({x}_{i}\right)=>m\displaystyle\int{\dfrac{\displaystyle\frac{d}{dt}\left({x}_{i}\right)}{g\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)}}d\left(\displaystyle\frac{d}{dt}\left({x}_{i}\right)\right)=\displaystyle\int{f\left({x}_{i}\right)}d\left({x}_{i}\right)$
Is this correct? What is the point of the exercise? Is there an analytical way to prove the equation of motion can't be integrated?
You can't integrate a) because in the the right hand side there is $f=f(x_i)$ and you've got a differential on $t$. If you rearrange terms, you can verify that $$m\frac{\mathrm d\dot{x_i}}{f(x_i)}=g(t)\,\mathrm dt$$ so that now you can't integrate the left hand side because $f$ depends un $x_i$ and you've got a differential on $\dot{x}_i$.
For b) you can integrate because each side has got the right variable $$m\frac{\mathrm d\dot{x_i}}{f(\dot x_i)}=g(t)\,\mathrm dt$$
For c) you cannot integrate because $f$ depends un $x_i$ and you've got a differential on $t$. $$m\frac{\mathrm d\dot{x_i}}{g(\dot x_i)}=f(x_i)\,\mathrm dt$$