This thought has occured to me a few days ago, and now I am puzzled about some fundamental properties /definitions in the theory of differential equations. Suppose I have an ode
$$\frac{dy}{dx}=y^{2}$$
In the above formulation, we are seeking $y$ as a function of $x$ so we have $x$ as the independent variable, $y$ is the dependent variable, and the ode is nonlinear autonomous ($x$ plays the role of time).
However if we take the reciprocal, we get $\frac{dx}{dy}=\frac{1}{y^{2}}$, and now $y$ is independent variable, $x$ is dependent, and the ode is linear and nonautonomous.
So given that we can interchange dependent and independent variables, what do these termns actually mean? Are they context-specific definitions, or is there some deeper intrinsic meaning? The same for linear/nonlinear and autonomous/nonautonomous?
Not only in the context of ODEs, in general, if you have a function $y=f(x)$ you can think of the inverse function $x=f^{-1}(y)$. In elementary calculus you have formulas for the inverse of a function, which are very useful. Note that, in some cases, the definition of a function as the inverse of another is the only definition we have at hand, e.g. the $\arcsin$, or even the exponential.
So the short answer is: yes, assuming a variable is dependent and the other is independent has only a physical meaning depending on the context.