Solving $P(x,y)dx + Q(x,y)dy =0$: interpretation in terms of forms I asked a similar question here which I will formulate more sharply: 
When we write a differential equation as $P(x,y)dx + Q(x,y)dy = 0$, what is the interpretation in terms of differential forms? 
(I suppose the language of differential forms is the proper one to understand it.) Suppose we can separate into $\alpha(x)dx + \beta(y)dy = 0$. We integrate to find a relation between $x$ and $y$. What is the interpretation of this action in terms of differential forms? At first I thought we were flowing along the vector field $\alpha(x)dx + \beta(y)dy$, and the relation between $x$ and $y$ describes the flow lines. But then I realized $\alpha(x)dx + \beta(y)dy$ is a covector field, not a vector field, so this interpretation is not correct.
When we integrate the right-hand side and get a constant, what is the justification of that in terms of forms? 

The  interpretation is the following: given a differential 1-form $\omega=Pdx+Qdy$ in the plane, you are asked to find its integral curves, i.e. 1-dim submanifolds of $\mathbb R^2$   whose tangent line at each point is annihilated by the 1-form. For example, the integral curves of $xdx+ydy$ are the concentric circles around  the origin. At a point where $\alpha\neq 0$ either $P$ or $Q$ do not vanish, hence  the same holds in a neighborhood of the point. If say $Q$ does not vanish, then  the integral curves are graphs of solutions to the ODE $dy/dx=-P/Q$. 
