Two variables in the definition of an ODE? Sometimes I see the definition of an ODE as a function of two variables $f(x,y)$, e.g.
$$
\frac{dy}{dx}=f(x,y)
$$
Does it mean $y=y(x,y)$?
Isn't an ODE always a function of one variable?
For instance here they use $y'=f(t,y)$. How should I interpt it?
 A: An example of a differential equation:
$$
\frac{dy}{dx} = x^2+y^2
$$
Of course $y$ is supposed to be a function of $x$ only.  In your general formulation, I took $f(x,y) = x^2+y^2$.  But the point of that
definition is that if you choose any function $f$ of two variables,
then you will
get an ODE.
note Your definition is only for an ODE of order $1$.  An ODE of order $2$ will be something like
$$
y''= g(x, y, y')
$$
Still, $y$ is a function of only one variable $x$,  But now $g$ is a function of three variables.
A: If $f$ is a variable of $2$ functions, it doesn't mean that those variables can't be dependent of one-another. In this case $f(x,y)=f(x,y(x))$. In other words, you can replace $f(x,y)$ with variable $x$ and a function $y$.
For example, if we look at an ODE
$$y'+y+x=0.$$
We can write it as
$$y'=-y-x\Rightarrow y'=f(x,y),$$
where $f(x,y)=-y-x$.
A: *

*Does it mean $y=y(x,y)$?
No, it doesn't. As pointed out we have for example the ODE $y'(x)=\cos(x)+y(x)$. That means on the right hand side we have some dependency on $x$ and $y(x)$. Therefore we write shortly $$y'=f(x,y)$$ where the first component indicates the variable of your ODE and in the second component we have the function $y$ that dependents on $x$. There are also ODEs of higher order, for example $y''(x)=y'(x)+y(x)+4x^2$. Then we write shortly $$y''=f(x,y,y')$$ where in the example we have $f(x,y,y')=y'+y+4x^2$. I think you get the idea. Generally for an $n$-th order ODE
$$y^{(n)}=f(x,y,y',y'',...,y^{(n-1)}).$$


*Isn't an ODE always a function of one variable?
Firstly an ODE is no function. It is a special type of equation I'd say. And as seen it also depends on the solution $y$ and its derivatives. But in some way you are right - the solution depends solely on one variable. If it depends on more variable we are in the setting of partial differential equations.


*For instance here they use $y'=f(t,y)$. How should I interpt it?
Don't be irritated by this. Now we just have $t$ as the variable. For example $$y'(t)=t^3+\sin(y(t)) \rightarrow y'=f(t,y)$$ where $f(t,y)=t^3+\sin(y)$. We can freely choose the variables, have a look at the differential equations
$$\begin{align} g'(h)=h^2+\ln(g(h)) &\rightarrow g'=f(h,g) \\ \theta''(\zeta)=e^{\zeta}+\theta(\zeta)+2\theta'(\zeta)&\rightarrow \theta''=f(\zeta,\theta,\theta') \\ \kappa'(\mathcal{O})=4\mathcal{O}+\kappa(\mathcal{O})^2 &\rightarrow \kappa'=f(\mathcal{O}, \kappa)\end{align}$$
