How should we think about equations like $dy = 2x \cdot dx$ from the viewpoint of modern geometry? We've just started learning about (smooth) manifolds at uni, and I'm kind of hoping this will finally help me get a handle on the dreaded Leibniz notation. Now I've read that expressions $dy$ like can be viewed as differential $1$-forms. I suppose this somehow allows us to make sense of equations like
$$dy = 2x \cdot dx$$
So I tried thinking through the details. Pretty soon, I decided it would be nice be able to write something like the following:

Let $M$ denote the smooth manifold given as follows.
Distinguished Projections: $y,x$
Equations: $y=x^2$
(The idea is that $M$ can be understood concretely as $\{(x,y) \in \mathbb{R}^2 \mid y=x^2\}.$)
Then $M \models y=x^2$.
Therefore $M \models dy = 2x dx$

I got thinking that maybe there is an algebraic component to all this. Perhaps we should be writing:

Let $M$ denote the smooth manifold presented as follows.
Generators: $y,x$
Relations: $y=x^2$
Then $M \models y=x^2$.
Therefore $M \models dy = 2x dx$

So I guess it would be nice if we could view smooth manifolds as the models of some kind of carefully chosen Lawvere theory. This couldn't possibly work, though, because relations like $y^2 = x^2$ don't yield well-defined smooth manifolds, because $0$ isn't a regular value of $x,y \mapsto y^2-x^2$.

Question. How should we think about equations like $dy = 2x \cdot dx$ from the viewpoint of modern geometry? Has it got to do with the
  models of some kind of a special Lawvere theory? And is there a useful
  generalization of the concept "smooth manifold" such that every
  equation involving smooth functions defines a smooth manifold, even equations like $x^2=y^2$?

I tried thinking about the Lawvere theory whose objects are $\{\mathbb{R}^n \mid n \in \mathbb{N}\}$ and whose arrows are smooth functions, but really wasn't able to get any insight into what the free algebras of this theory look like. (Does anyone know?)
 A: 1) The equation $dy = 2x \cdot dx$ is an equality between two sections of the cotangent bundle $T^*M$ to the manifold $M$.
It means that at any point  $m=(a,b)\in M$ and for any tangent vector $u\frac {\partial}{\partial x}|_m+v \frac {\partial}{\partial y}|_m\in T_mM$ we have $v=2a\cdot u$.
It is of course to be expected that there must exist  a relation between $dy$ and $dx$ since the cotangent vector space $T^*_m M$ to $M$ at $m$ is $1$-dimensional, so that there must be a linear relation between $dy$ and $dx$.
Note carefully that in the above $x,y$ are functions  $M\to \mathbb R$, namely the restrictions to $M$ of the  two coordinates on $\mathbb R^2$.  
2) The generalization of the concept "manifold" to equations of the form $x^2=y^2$ exists if you restrict your functions to polynomials or rational functions : these objects are called real algebraic varieties and have been intensely studied.
There have been attempts to extend these to differential varieties, defined by arbitrary differentiable functions, but this is a very specialized area of differential geometry. 
3) To my knowledge Lawvere theory has had zero impact on differential geometry: I am not aware of  any serious result in that field  ever being obtained by using it.  
