I'm reading Bott & Tu's Differential Forms in Algebraic Topology and its Exercise 1.7 asks me to compute $H_{DR}^*(\mathbb{R}^2-P-Q)$ where $P$ and $Q$ are two points in $\mathbb{R}^2$. I've figured out how to compute this for degree $0$ and $1$, but get stucked on showing that $H_{DR}^2(\mathbb{R}^2-P-Q)=0$. I know how to show this using Mayer-Vietoris sequence, but this exercise is put just before the section of Mayer-Vietoris sequence, so I really wonder a solution without the use of that.
I attempted to show that every $2$-form $f(x,y)dxdy$ is exact using the integration on path to find $\alpha dx+\beta dy$ such that $d(\alpha dx+\beta dy)=fdxdy$, but it seems that with two holes in $\mathbb{R}^2$ there is not a legal choice of pathes that may work.
To find $\alpha$ and $\beta$ is exactly to solve the differential equation $\frac{\partial \alpha}{\partial y}= -\frac{f}{2} $ and $\frac{\partial \beta}{\partial x}=\frac{f}{2}$. Is there any result for this in PDE?
Thanks in advance for any hint, solution or reference that contains a solution.