[Physics] Integrable equations of motion

homework-and-exercisesmathematical physics

Suppose that a force acting on a particle is factorable into one of the following forms:

$$\text{a)}\,\,F(x_{i},t)=f(x_i)g(t)\,\,\,\,\,\,\,\text{b)}\,\,F(\dot{x}_{i},t)=f(\dot{x}_{i})g(t)\,\,\,\,\,\,\text{c)}\,\,F(x_{i},\dot{x}_{i})=f(x_i)g(\dot{x}_i)$$

for which cases are the equations of motion integrable?

I know the answer is b, as this is an example. However, I don't feel that clear on why the other two aren't integrable. For instance,

$$\text{c)}\,\,m\frac{d\dot{x}_{i}}{dt}=f(x_i)g(\dot{x}_i)$$

if I do some manipulation with this equation…

$$m\frac{d\dot{x}_{i}}{g(\dot{x}_{i})}=\frac{f(x)}{m}dt$$

Why is that false for integrability?

You can't integrate the right hand side because $f=f(x_i)$ and you've got a differential on $t$. As for a), if you rearrange terms, you can verify that $$m\frac{d\dot{x_i}}{f(x_i)}=g(t)\,dt$$ so that now you can't integrate the left hand side because $f$ depends un $x_i$ and you've got a differential on $\dot{x}_i$.