In the Cartesian plane, $\mathbb{R}^2$, we have a rectangular domain $\Omega \subset \mathbb{R}^2$, with $x\in[0,a], y\in[0,b]$, where $a,b\in\mathbb{R}^{+}$. Over a subinterval $x\in[x_{1},x_{2}]\subset [0,a]$, we have two arbitrary curves (or two families of curves) defined as $y_{1}(x)$ and $y_{2}(x)$.
We need to find the family of curves $y_{1}(x)$ and $y_{2}(x)$ that satisfy the condition (if not exact zero, then approaching it):
$$ \int^{x_{2}}_{x_{1}} \left[y_{1}(x)-y_{2}(x)\right]\cos\frac{2\pi x}{a}dx \rightarrow0 $$
How can we proceed to find the form of such curves ($y_{1},y_{2}$)? Even if we could assume the shape of one curve (e.g. say taking one of them as a specific known equation), how can we generally find the other curve?
Any insight/tips would be appreciated. Also, it would be useful to know the subbranches of applied math (or related books/literature) concerned with similar problems.