I'm currently reading Chapter 7 from Lectures on Symplectic Geometry by Da Silva.
Here's a linked question.
Let M be a compact manifold and let $w_0$, $w_1$ are symplectic forms on M. If $[w_t]$, $0\leq t \leq 1$, is a smooth family of closed 2-forms joining $w_0$ to $w_1$ s.t. $\frac{d}{dt}[w_t]=[\frac{d}{dt}w_t]=0$, there exists a SMOOTH family of 1-forms s.t. $$\frac{dw_t}{dt}=d\mu_t, 0\leq t \leq 1.$$
Existence of a family of 2-forms seems natural, but the author points out that in order to show that there exists a SMOOTH family, one uses the Poincare lemma for compactly supported forms and Mayer Vietoris sequence in order to use induction on the number of charts in a good cover of $M$. But I'm having trouble showing the existence. Any help would be appreciated!