Given a projective non-singular curve $X$, the a Serre duality asserts an isomorphism between $H^0(X,\Omega^1_X)$ and dual of $H^1(X,\mathcal{O}_X)$. My question is how to compute the dual elements in concrete examples.
For instance, it is well known that if the curve $X$ is given by affine equation $y^n=x(x-1)(x-\lambda)$, then $H^0(X,\Omega^1_X)$ is generated by the forms, $dx/y^i$ for $i=\lceil \frac{n+1}{3}\rceil,...,n-1$ and $xdx/y^i$ for $i=\lceil \frac{2n+1}{3}\rceil,...,n-1$. How to compute the corresponding classes in $H^1(X,\mathcal{O}_X)$ in this example?