The definition of an ordinary differential equation is $$ F[x; y(x),y'(x), \cdots, y^{n}(x)]=0 $$ I don't understand the meaning and notation of the definition. What does $F$ mean?
Suppose I have a first order ODE, $n=1$, so $F[x; y(x),y'(x)]=0$: \begin{align} y'(x) &= \frac{1}{x} \\ \frac{dy}{dx}&=\frac{1}{x} \\ y(x)&= \log\lvert x \rvert+C \end{align} In this exemple, what does $F$ and $F[x; y(x),y'(x)]=0$ mean?
Or if I have \begin{equation} s'(t)=\cos 2t ,\qquad s(t)=\frac{\sin 2t}{2}+C \end{equation} How can I interpret the definition here?
$F(x,y(x),y'(x),\dots)$ means a function (a special relation) of the variables $x, y(x), y'(x)$. So it could be any equation involving these variables. In your first example, $F(x, y'(x))=y'(x)-1/x$. In your second example, $F(t, s'(t))= s'(t)-\cos 2t$. So an ODE gives you a relation of $x, y(x)$, and $y'(x)$, (or with higher order derivatives).