How to interpret $dx$ on its own in differential equations? I have come across some variations in the notation of differential equations that seem to be so obvious to everybody that nobody cares to explain the meaning properly.
There is a notation version that I understand like e.g. this one representing angular change in a pendulum
$$\ddot{x}(t) = - \mu \dot{x}(t) -\frac{g}{L} \sin(x(t))$$
which should be the same as
$$\frac{d^2x}{{dt}^2}(t) = - \mu \frac{dx}{dt}(t) -\frac{g}{L} \sin(x(t))$$
where I understand $\frac{dx}{dt}(t)$ as in the rigorous analysis definition
$$\frac{dx}{dt}(t) := \lim_{h \to 0}\frac{x(t+h)- x(t)}{h}$$
To me $\frac{dx}{dt}$ was until now a kind of mnemonic representation for the idea that we look at the tiny changes in the function output given a tiny change in function input and see what happens when that tiny input change goes to $0$. Now I see differential equations like this one defining the Ornstein-Uhlenbeck process:
$$dX_t =  \theta \cdot ( \mu - X_t) dt + \sigma dW_t$$
Of course this is different in a variety of ways but my question is with regards to the use of $dX_t$ or $dt$. In the first equation the $dx$'s and $dt$'s seemed to occur only together, representing the rigorous definition of the derivative via a limit. Here they seem to have a life of their own and a separate meaning.
How can one (1) interpret and (2) handle terms like $dX_t$ or $dt$ arithmetically when they stand alone like this?
 A: The point is that in a certain sense $dW_t$ is the square root of $dt$, which makes any idea of differential quotients quite doubtful.
The standard understanding is that this equation of increments is purely symbolic, it gives a frame for the coefficients of $dt$ and $dW_t$. As a proper equation, the symbolic equation $dX=adt+bdW$ stands for the integral equation
$$
X_t=X_0+\int_0^ta_sds+\int_0^tb_sdW_s
$$
where the first integral is the usual integration of continuous (with some caveats) functions, while the second is the Ito integral that is specifically defined on the basis of the Brownian motion and stochastic processes that similar enough to it.
This single integral equation is also equivalent to the family of integral equations
$$
\int_0^tg_sdX_s=\int_0^tg_sa_sds+\int_0^tg_sb_sdW_s
$$
where again $g$ is a continuous function or some slight generalization.
As one can see, the possible manipulations do not "care" about the integral sign, which is why it is often left out of the notation of such equations and their transformations.
