Basic question about terminology, notation and definitions in calculus When reading stuff about differential equations I'm coming across some strange (for me) notations/terminology. For example, when coming across something like this:
$$\frac{dy}{dt}=f(y,t)$$
or
$$y'=f(y,t)$$
$y$ and $t$ are referred to as variables. However, eventually during solution of the differential equation, at a seemingly arbitrary point, the notation suddenly switches to 
$$y(t)=f(t)$$
Where $f$ is now a function of $t$. Now intuitively it's quite clear that this whole time $y$ has been regarded as being dependent upon $t$, so can be though of as being a function of $t$. However, what exactly justifies talking of $y$ as a variable at some points, while talking of it as a function at other points.
Or are we talking about the 'function variable' $y$? But then why do we only sometimes make it explicit that it is a function in $t$ (by writing $y(t)$) while at other times omitting that information (by just writing $y$).
 A: In the equation $x=2x-1$, $x$ is a variable, yet it's really just equal to $1$. So up until it's being solved for, it's a variable. In your case, you're solving for $y$, a variable function of a differential equation. $y$ is not a number, it's a function of $t$. Suppressing $y(t)$ to $y$ just makes notation cleaner because for some reason remembering $y$ depends on $t$ is easier than writing out $y(t)$ at every step. 
When solving differential equations, it's often cleaner (and faster to write) having short notation like $y'$ instead of $\frac{dy}{dt}$. The important thing is to remember what the prime refers to, in this case being differentiation with respect to $t$. At the end, it's usually polite to review what the function depends on, especially if you're about to start plugging in boundary conditions. 
So the short answer is, its a minor abuse of notation which is usually clear from the context. When it's not clear, it helps to point it out. In the answer, write $y(t)$ to emphasize dependence. 
A: The idea here is that we have a function $y$ that depends on a variable $t$,  so in that sense it's just your basic calc 1 $y(t)$.   However, we have a differential equation that tells us that the derivative of $y$ with respect to $t$ is itself equal to a function that can depend on both $y$ and $t$. 
For example,  we could have $\frac {dy} {dt}=t^2$, here we just have the derivative is $t^2$, which means you can treat this like a calc 1 problem.
However, we might have the derivative itself is related to the VALUE of the function at a given time in addition to the time.  For example,  
$\frac {dy} {dt}=y\cdot t$.
This says that at any given point, we can get the derivative of $y$ with respect to $t$ by multiplying the input and the output (i.e. $y(t)\cdot t)$)
In this case, the function on the right hand side would be $f(y,t)=y\cdot t$,  and we would have the diff eq as $y'=f(y,t)$
A: We say $y$ is a "function" of another variable when we are interested in the behavior of its value in relation to the value of another variable; but we say $y$ is a "variable" when we are interested in the value itself, or when we are talking about some other function that is dependent on it.  Mathematicians endeavor to be precise with the use of language but the reason why we resort to equations and symbols is essentially because the vernacular frequently lacks the needed precision.
