Meaning of $\frac{d}{dt} x(t)=f(x)$? If an ODE is written as $\frac{d}{dt} x(t)=f(x)$, does it mean $$\frac{d}{dt} x(t)=f(x(t))$$ or $$\frac{d}{dt} x(t)=f(t,x(t))$$ or maybe something else? 
What is the difference between the above equations?
 A: The only difference is that in the first case, $f$ is not permitted to explicitly depend on $t$. For example, if $x(t) = t^2$, then $f(x(t))$ shouldn't be able to "tell" whether it's been given $t$ or $-t$, since $x$ will be the same in either case. In the second case, $f$ can appeal to either $t$ or $x$ freely.
For example, $\frac{d}{dt}x(t) = x(t)^2$ is an example of the first case. $\frac{d}{dt}x(t) = t + x(t)^2$ is an example of the second case, since it appeals to both $t$ and $x(t)$. However, in the second case $f$ doesn't have to use $t$ if it doesn't want to; $\frac{d}{dt}x(t) = x(t)^2$ is an example of the second version, too.
Your original equation, $\frac{d}{dt}x(t) = f(x)$, is of the first form; the notation "$f(x)$" means that the only variable $f$ may draw on is $x$, not $t$. $\frac{d}{dt}x(t) = f(x)$ is of the form $\frac{d}{dt}x(t) = f(t, x(t))$, but it carries the additional information that $t$ itself is not used.
A: The first equation is correct.
$$\dfrac d {dt} x(t) = f(x(t)) $$
It looks like the equation is from kinematics. $x$ is dependent variable, i.e. its value depends on $t$, and the function to find the value of $x$ from $t$ is written here as $x(t)$. It's a single variable function, it can be written only in terms of $t$. Therefore, $\dfrac d {dt} x(t)  $ will also be in terms of $t$. Let $ \dfrac d {dt} x(t) = g(t) $. What the equation in your title implies is that $g(t)$ can be writen in terms of $x$, as $f(x)$, using the function $x(t)$. 
Example:
Let,  $x(t) = e^{2t} \implies  \dfrac d {dt}  x(t) = 2e^{2t} = 2x(t) = 2x $
Here, $f(x) = 2x $ and $f(x(t)) = 2x(t) = 2e^{2t}$
The second equation you posted is a valid one, but not what the equation in your title implies. 
Example:
Let $x(t) = e^{t^2} \implies  \dfrac d {dt}  x(t) = 2te^{t^2} = 2t(x(t)) $, which is a function of t and x(t). 
